What are two symbolic techniques used to solve linear equations? Which do you feel is better? Explain why. • Post an example for your class to solve.

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There are more than two techniques used to solve linear equations, but two of the different methods that I chose to solve with are by using graphing, and by using elimination. Graphing can be used by solving for the intercepts of the equation, by solving with x equal to zero, and with y equal to zero. This is the quickest way to find the x and y intercepts, and draw a line in-between them. Another method is to make sure that the represented b in this equation is the y intercept. The slope is the m in the equation. The second method is to solve by elimination. This method makes it so that either the x or y variable in the two linear equations can be multiplied to equal the variable in the other equation. This makes one of the variables equal to the other side, and then can be combined to solve for the second variable. The value that you come up with is then plugged into the original equation and then solved for the other variable. A third method is to solve by using substitution. This method sets one equation equal to a single variable such as y, then plugging that value in for the y in the other equation. This method is the most difficult. I prefer solving using graphing. This is how I would most easily solve two linear equations. My Classmates can solve this problem: 3x+4=y, and 4x-6=y

Do the equations x = 4y + 1 and x = 4y – 1 have the same solution? How might you explain your answer to someone who has not learned algebra?

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They do not have the same solution simply because you can’t add and subtract a number from the same equation and get the same number. It just doesn’t work, especially in a line graph. If I was to explain this to someone who didn’t know what algebra was I would first take away the x value and just leave it blank. I would then assign a number to y, like 3. I would multiply it out, and show them that when 4y+1 = 13 it is not the same answer as when 4y-1 = 11. This would probably be the easiest way to explain that adding and subtracting that same number isn’t going to give you a similar value. If they still didn’t understand what I meant by the explanation I have used, I would then substitute 4 in for the 4y. I would ask them if 4+1 is the same as 4-1. I would illustrate perhaps by saying would you like 4 dollars plus 1 dollar or 4 dollars minus 1 dollar. Any logical person will choose to take the 4+1 dollars and more simply understand that the x value represented in the equation is just the answer after adding or subtracting the 1.

How many solution sets do systems of linear inequalities have? Do solutions to systems of linear inequalities need to satisfy both inequalities? In what case might they not?

There can be an infinite number of solutions to a linear inequality. The only real case where there is one solution is when the inequality has two cases and they are both and such as: x ≤ 3 AND x ≥ 3. In this inequality there is only one answer and that answer is 3. Solutions to systems of linear inequalities usually need to satisfy the condition of both inequalities if they are connected by an

AND. In cases where they only need to satisfy one of the inequalities is when they are connected by an OR. For instance in the case x < 3 OR x > 2, there is any number of solutions to this problem. In fact in that particular problem any real number would satisfy the conditions of the line graph. In the equation x < 2 AND x > 3, there is no real solutions because no number can satisfy the conditions of both inequalities.

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Provide one real-world example of when graphing could be useful. Do you think you would ever use graphing in your life to solve problems? Explain why or why not.

Graphing is useful in everyday life. I use graphing in my business to track the amounts of hits that I get on my website, as well as the amount of traffic that is generated from Google or Bing. I use graphing currently in my everyday life and I find it essential to my business. Graphing is something that is essential to many different forms of business and must be used on a daily basis. It makes everyday life simpler, and easier to understand predictions throughout time. Graphs are used for literally any type of data, and can help to track any historical progress throughout a varying time period. The easiest way to track I feel is to make your time variable the x variable, any other stats that you may be wanting to track your y variable.

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What concept learned in this course was the easiest for you to grasp? Why do you think it was easy for you? Which was the hardest? What would have made it easier to learn?

The concept in this class that was the easiest to grasp was definitely the standard form of the graphing line. I feel that the explanation of y=mx+b is the simplest way to describe a line. It is so simple in that b will always be the y intercept, and m will always be the slope which is the rise over the run. If the slope is negative, you can go down and over or up and back it is a constant piece of math which will never change and is one of the most useful tools for daily business endeavors. The hardest thing for me to graph is the inequalities of a line. It is very difficult to grasp the concept of flipping a sign when it needs to be flipped. I think through the DQ’s in this class that I have been able to learn a lot as time has gone on, and the concept of inequalities has been made easier to learn. I think that the Dq’s are the most helpful when it comes to learning new material.

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Week 2 Concept Check Post your 50-word response to the following: • How do you know when an equation has infinitely many solutions? • How do you know when an equation has no solution?

An equation has an infinite number of solutions when the variable equals itself. For example, x=x, both sides are equal. No matter what values are given for x the terms of the equation will always be met. If an equation has no solution you will get an untrue answer such as 6=2 when you see something like 6=2, it has no solution because 6 cannot equal 2. Only 6=6 and 2=2.

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Exercise: Week 4 Concept Check Post your 50-word response to the following: Explain in your own words why the line x = 4 is a vertical line.

If the value of x=4 then no matter what the value of y is the value of x will always be 4. When you draw an x and y axis graph, each point on the line represents the place where an x and y value meet. x=4 means that, no matter what the value of y, the x value will always be 4. Every y value on the line meets the matching x value at 4, resulting in a vertical line.

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Exercise: Week Six Concept Check Post your 50-word response to the following: How can you determine if two lines are perpendicular? Post in your Individual Forum

Non vertical lines are perpendicular if the product of the slopes equal -1. So essentially, (M1) (M2) will equal -1. Also if you can make a little box (or 90 degree angle) then lines will are perpendicular. If one line is horizontal and another is vertical they will also be perpendicular.

How do you think you will use the information you learned in this course in the future? Which concepts will be most important to you? Which will be least important? Explain your answers.

Math is a very useful tool. It has many practical applications from basic graphing all the way to complex equations involving physics and many other things. You can use it to determine the family income needs as well as professional or job related tasks. Math can be used to track statistics for literally anything.

            In my job I am able to track through the internet demographically who buys what products. This is something that I can track with a simple scatter graph. In my personal life I can use some of the geometric formulas I learned to determine what the layout and required materials I need for my hobbies such as sewing.

            I probably will not use some of the more intricate formulas such as solving inequalities. These types of formulas simply do not serve a purpose in my job or personal life. But I believe that it is good to know these formulas and problem solving skills. 

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Using the readings discussed in this course, provide one real-world application of the information learned that has been the most valuable to you. Why has it been valuable?

Learning about the differences between scatter plots and line graphs has been some of the most practical information that I have learned during this class. Scatter plots express data in such a way that any amount of random data can be shown and relationships can be drawn in one graph. Scatter plots can be used practically to show whether or not a group of people meet certain criteria.  Line graphs express data so that a difference, or slope, can be shown from period to period. Line graphs can be used to express information from year to year, or week to week to show improvements so that values can be tracked. Different information can be classified through different types of graphs, this is why line and scatter graphs are useful in real-world situations.

What concept learned in this course was the easiest for you to grasp? Why do you think it was easy for you? Which was the hardest? What would have made it easier to learn?

The easiest concept to grasp in this math class was the explanation used to determine when to flip the sign in an inequality. Whenever you have to divide or multiply by a negative number the sign in an inequality changes or flips. This is the easiest explanation of when to flip the sign of inequalities; I was never sure when I studied the topic in high school. One of the hardest concepts to grasp was the use of PEMDAS. I never understood why there must be order to solving equations throughout the equation. It was difficult for me to grasp the order, but once I read up on the topic I understood that there must be an order so that regardless of the writing of the equation it will always be solved the same way. It would have been easier to understand if the concept that it is just a certain order to solve math all of the same way was explained. 

Using the readings discussed in this course, provide one real-world application of the information learned that has been the most valuable or meaningful to you. Why has it been valuable?

Learning about the differences between scatter plots and line graphs has been some of the most practical information that I have learned during this class. Scatter plots express data in such a way that any amount of random data can be shown and relationships can be drawn in one graph. Scatter plots can be used practically to show whether or not a group of people meet certain criteria.  Line graphs express data so that a difference, or slope, can be shown from period to period. Line graphs can be used to express information from year to year, or week to week to show improvements so that values can be tracked. Different information can be classified through different types of graphs, this is why line and scatter graphs are useful in real-world situations.

How many solution sets do systems of linear inequalities have? Do Solutions to systems of linear inequalities need to satisfy both inequalities? In What case might they not?

A solution set may have any number of solutions, the most common answers for this are; 0,1 or infinite. An example where only one is had is when x≤3 and x≥3. An example of a solution where none are found is when given x>3 and x<2. Both of these cannot be satisfied at the same time, so there is no solution set for this number. If a system does not fit into both of the inequalities and the word “and” is used, there will be no solutions to the given problem. The last option is when you are given something like x>3 and x>4. This is a solution where any number greater than 4 will satisfy the conditions, and will be considered infinite. 

What are two symbolic techniques used to solve linear equations? Which do you feel is better? Explain why. Please be sure that you pay attention to the bolded word. Be specific in your explanation and use examples to support your response:

Two symbolic techniques used to solve linear equations are; graphically, and algebraically. Personally I prefer to use graphically the most because it is the fastest method to solve the equation, also it is easy to solve with a calculator.

To solve an equation graphically you must first set it into slope intercept form. This is achieved by setting the equation equal to Y. In this given problem, x + y = 10, and 3x+2y=20 we will solve for y. You get -x+10 = y and -3/2x+10=y. When graphing these two numbers check to see where they intersect, which is at (0,10). Where they intercept is the answer to the problem.

The second method to solve by is algebraically. This method uses substitution and can be a bit confusing if a clear paper trail is not kept when doing your homework. Given the same problem, x+y=10, and 3x+2y=20 we will take the first half of the problem and set it all equal to y. We get –x+10=y. Given this information, take the y value and substitute it into the other problem (3x+2y=20). Substituted we get, (3x-2x+20=20) Simplified the equation comes out to (x+20=20). Therefore the value of x is 0. We then take this value of x and substitute it into the first half of the 


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linear equation. Where (x+y=10) when substituted this turns into (0+y=10) Therefore the value of Y is 10. Given this information we can conclude that the answer to this linear equation is (0,10) (x,y).

A third valuable method is to solve by elimination. Given the same problem, x+y=10, and 3x+2y=20, we will solve by eliminating the variable x. The goal here is to make it so that the x values will cancel out when the problem is combined together. The first step is to multiply the first linear equation by -3. This is done here -3(x+y=10) we get, -3x-3y=-30. This new linear equation is combined with the second one, (3x+2y=20). Together when combined This new equation comes out to (3x-3x+2y-3y=-10) When canceled out it becomes, -y=-10. y therefore equals 10. We plug this value back into either one of the original linear equations, and can conclude that x is again 0.

In conclusion I feel that the first method, solving graphically, is a far faster method to solve the problem. Not only that, but you can see just by the explanation of the two types that substituting and solving algebraically takes longer to explain, and longer to follow through with. 

MAT 116 Week 8 Exercise Concept Check

A square bracket means that the interval includes that endpoint. Since in this problem both brackets are square, both endpoints are included. This interval goes from -4 to 10 and includes the endpoints. On a number line, one can fill in those endpoints. Alternatively, instead of filling in the endpoints, one can draw a bracket at -4 and another bracket at 10. If the interval had not included parentheses, one would not have filled in the circles (or if one used brackets/parenthesis, he or she would have put a parenthesis on the number line). A straight line and mark a line from -4 to 10 with [at -4 and] at 10

Give an example of a division of a polynomial long division by a binomial and show all the steps when performing the division. The example must be your own and not from the text book. How is this similar to numerical division of real numbers? Give another

Polynomial division is very similar to long division that we learned in elementary school. Instead of simply using numbers, there are variables as well. The process, though, is effectively the same. We go through the problem term by term, just like in standard numerical long division. If we understand how to do one type of long division, it is quite easy to extend the technique to the other type of division. As long as we know how to multiply monomial terms with variables, the actual process is the same. The polynomial goes to the inside of the division symbol, and the binomial goes outside. We try to get the first term of polynomial from the binomial first term, and repeat this till we are able to get a polynomial of lesser degree than binomial (remainder) or till we get zero remainder. Example (x^2 + 5x + 6) divided by x + 5 We get X+5 /x^2 + 5x + 6 First we multiply by x to get x^2 + 5x We get X + 5/ x^2 + 5x + 6 -x^2 – 5x 6 We get the remainder as 6 Here quotient is x and the remainder is 6 If we divide same polynomial by x + 2 We get X+2 /x^2 + 5x + 6 First we multiply by x to get x^2 +2x, and then by 3 to get x + 6 We get X+5/ x^2 + 5x + 6 -x^2 – 2x X + 6 -x – 6 Zero remainder The quotient is x + 3 and remainder 0

Thread for Week 3 - Discussion Question #1 Take any number (except for 1). Square that number and then subtract one. Divide by one less than your original number. Now subtract your original number. Did you reached 1 for an answer? You should have. How do

Suppose we take the number 5 Squaring 5^2 = 5*5 = 25 Subtracting 1 25 -1 = 24 Dividing by 1 less means 5-1 = 4 24/4 = 6 Subtracting original number 6 – 5 = 1 Yes answer is 1. Taking variable y Squaring y^2 Subtracting 1 y^2 -1 Dividing by 1 less means y^2-1 / (y-1) = (y+1)(y-1)/(y-1) = y+1 Subtracting original number y+1 - y = 1 Yes answer is 1.   Here is a number game that uses the skills of simplifying rational expressions. Take any number (except for -4) and add 2. Next, multiply by 2 less than the number. Add 3 times the original number. Divide by 4 more than the original number. Finally, add 1. You should be back where you started! Here it is with symbols: Take any number (except for -4) and add 2. x x+2 Next, multiply by 2 less than the number. (x+2)(x-2) = x^2 - 4 Add 3 times the original number. x^2 + 3x - 4 = (x+4)(x-1) Divide by 4 more than the original number. (x+4)(x-1)/(x+4) = x-1 Finally, add 1. You should be back where you started!

Give an example of a sum of two rational expressions with different denominators, then perform the operation by showing all the steps, including how you found the common denominator. These rational expressions must have a variable in the denominator, such

Rational expression with different denominators. x/(x + 2) + x-2 /( x+3) we find the LCD which is (x+2)(x+3) here making these like fractions.. so that we can add x(x+3)/(x+2)(x+3) + (x-2)(x+2) / (x+3)(x+2) x^2 + 3x / (x+2)(x+3) + x^2 – 4 / (x+2)(x+3) x^2 + 3x + x^2 – 4 / (x+2)(x+3) 2x^2 + 3x – 4 / (x+2)(x+3) LCD is found by finding the factors of the denominators just as we do in the case of fractions. Example for classmates… Add x – 2 / x^2 – 4 + (x + 3) / (x – 2) Rational Expression can be used in real life where we need to solve for questions which involve fractions. Example: Speed, Distance Cases, or Time and work Cases. Here is a number game that uses the skills of simplifying rational expressions. Take any number (except for -4) and add 2. Next, multiply by 2 less than the number. Add 3 times the original number. Divide by 4 more than the original number. Finally, add 1. You should be back where you started! Here it is with symbols: Take any number (except for -4) and add 2. x x+2 Next, multiply by 2 less than the number. (x+2)(x-2) = x^2 - 4 Add 3 times the original number. x^2 + 3x - 4 = (x+4)(x-1) Divide by 4 more than the original number. (x+4)(x-1)/(x+4) = x-1 Finally, add 1. You should be back where you started! x

Why is it important to simplify radical expressions before adding or subtracting? How is adding radical expressions similar to adding polynomial expressions? How is it different? Provide a radical expression which consists of a sum or difference of two ra

Why is it important to simplify radical expressions before adding or subtracting? It is important to get radical expressions into their simplest form before you add or subtract them. You can only add and subtract like radicals (for example ). You cannot add them if they are different (for example ). How is adding radical expressions similar to adding polynomial expressions? How is it different? It is similar because in polynomial expressions, you can only add like terms. In radical expressions, you can only add like radicals. They are different, though, because instead of powers, you are dealing with roots. Provide a radical expression for your classmates to simplify. Here is one to simplify: The Answer is

Review section 10.2 (p. 692) of your text. Describe two laws of exponents and provide an example illustrating each law. Explain how to simplify your expression. How do the laws work with rational exponents? Provide the class with a third expression to sim

Describe two laws of exponents and provide an example illustrating each law. -In multiplying, we can add the exponents as long as the base is the same ex. 3^2 x 3^4= 3^2+4 -To raise a power to a power, multiply the exponents ex. (3^2)^4= 3^2*4 Explain how to simplify your expression. -Convert to an exponential expression, use math to simplify the exponent, and convert back to radical expression when appropriate How do the laws work with rational exponents? -When using radical exponents, simplify the exponent after coverting to exponential exression, rather than simplifying the expression. Provide the class with a third expression to simplify that includes rational (fractional) exponents. 4^2/3 *4^1/3

How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain. Provide your classm

How do you know if a quadratic equation will have one, two, or no solutions? We calculate the discriminant, b^2-4ac. If it's negative, there are no solutions. If it's positive, there are two solutions. If it's zero, there is one solution. How do you find a quadratic equation if you are only given the solution? If we are given solutions a and b, we plug them into (x-a)(x-b)=0 and the required equation is x^2 - (a+b)x + ab = 0 Is it possible to have different quadratic equations with the same solution? Explain. Yes, if they are multiples of each other, they'll have the same solution(s). For example, x^2 and 2x^2 have the same solution. Provide your classmates with one or two solutions with which they must create a quadratic equation. Solution: -3 , -4 Answer: (x-(-3))(x-(-4))=0 (x + 3)(x + 4)=0 x^2 + 7x + 12 = 0

How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain. Provide your classm

How do you know if a quadratic equation will have one, two, or no solutions? We calculate the discriminant, b^2-4ac. If it's negative, there are no solutions. If it's positive, there are two solutions. If it's zero, there is one solution. How do you find a quadratic equation if you are only given the solution? If we are given solutions a and b, we plug them into (x-a)(x-b)=0 and the required equation is x^2 - (a+b)x + ab = 0 Is it possible to have different quadratic equations with the same solution? Explain. Yes, if they are multiples of each other, they'll have the same solution(s). For example, x^2 and 2x^2 have the same solution. Provide your classmates with one or two solutions with which they must create a quadratic equation. Solution: -3 , -4 Answer: (x-(-3))(x-(-4))=0 (x + 3)(x + 4)=0 x^2 + 7x + 12 = 0

Quadratic equations may be solved by graphing, using the quadratic formula, completing the square, and factoring. What are the pros and cons of each of these methods? When might each method be most appropriate? Which method do you prefer? Explain why.

The graphing method allows you to visualize quadratic equation solutions, since the solution(s) to the equation occur where the graph intersects the x axis. But if you aren't given a graph, this method is tough, since it's tough to come up with an accurate graph. This method doesn't help if the solutions are imaginary. The quadratic formula is a good way to get the answer regardless of the equation. If you can remember the formula, then you can solve any equation you are given, even if the solutions are imaginary. Completing the square is similar, in that if you remember the process, you can always get the answer. Completing the Square can be used to solve any quadratic equation. It also helps in graphing quadratic equations. Some of the cons for this method also involves more steps and can seem a lot more complicated. It is also slower the using the quadratic formula. When the equation is in the form x^2=d, or (x+c)^2=d, or when creating a binomial square is straightforward, then use this method. I think that factoring is the quickest and easiest method, but it only works for "simple" equations that have "nice" solutions. If you aren't good at seeing the factors, this can be very hard. I would use the graphing method if I am already given a graph of the quadratic. The formula and completing the square are useful for equations with more complex (or imaginary) solutions that aren't integers or simple fractions. Factoring is a good way to solve equations with whole number solutions. I prefer to use the factoring method, if I can, since it allows me to come up with the solutions in one easy step. Each of the other methods take longer.

Describe in details two concepts investigated in this course (not in MAT 116) that can be applied to solve real life problems in general and your personal and professional life. In what ways did you use MyMathLab® or the Center for Mathematic Excellence f

The content in this course has allowed me to think of math of a useful tool. I have really learned a lot of math and grasped it more in this class. The content in this course showed me how math is used in everyday life. Although for myself I do not use math that often but I can see the benefit for many. For instance, using radical expressions to solve for answers such as bisecting an area of land or section of a project and using rational expressions to find estimates on such things as interest payments. I used the MyMathLab as a study guide for practice problems and to help prepare myself for the quizzes and tests, and final. This guide allowed me to prepare and take on my anxiety of quizzes. Using the guide, allowed me to study at my own pace and understand the material easier. Evaluate the radical expression when a = 2 and b = 4. Choices: A. 9 B. 8 C. 7 D. 6 Correct Answer: D Solution: Step 1: [Substitute the values of a and b in the given radical expression.] Step 2: [Find the positive square root.] Step 3: [Multiply.] Step 4: [Add.] Step 5: = 6 [Simplify.]

What are the practical uses of scientific notation? Why is scientific notation so important in modern-day society?

The practical use of scientific notation is when you have a large number and you need to make it into a smaller number. When you have a number like 8 trillion it is easy to express that by condensing the number and making it easier for one to understand. Practically it is very useful in chemistry where large fractions can be present. It can be helpful when you need to keep track of these large numbers and ensure that you don’t miscount when you look at the number of zeroes in an equation.

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Using the readings discussed in this course, provide one real-world application of the information learned that has been the most valuable to you. Why has it been valuable?

The most well used real world application to me has been the implementation of graphs and the slopes that dictate them. These are helpful to me in my job because they help me to track the progress of client satisfaction from month to month. It is a very useful way to hundreds if not thousands of different stats in sports. It can be useful throughout all types of work and be used as a tool to help accomplish all kinds of work. The second part of math that I have enjoyed is the part where we learn how to use graphs to solve word problems. These can be very informative and help you learn how to use the classic math formula to solve problems.

If your son or daughter asked you why they needed to learn math in school, what would you tell them?

I would simply tell my son or daughter that I need math in my everyday life so it is certain that they too will need math in their life. I would try and explain to them the simple value of money and explain that they need to know how to do math to use their money right. This is a common thing that they will encounter throughout life and they will need it each and every day of their life. It is important that they succeed in math and try their hardest to understand it because a brain that understands math well is a sharp brain that can comprehend some of life’s greatest problems. It helps to keep your brain healthy when you use math to keep it actively engaged. Math is a tool that is a necessary part of life, and in order to succeed in life it is essential to have a solid grasp on math’s basic foundations.

Knowing what you know now about mathematics, how would you explain to a friend the value of mathematics in everyday life?

Math is valuable whether it be in visiting the grocery store or compounding the monthly or yearly interest rate on the new car that you purchased. Math is an invaluable tool that is used when completing even the most mundane tasks. It is essential that an individual understands math to be successful in life. I use math in my everyday life whether that be calculating how much I have left in my bank account or how much I can spend on my groceries. I may use it in calculating the amount of interest I still owe on my student loans or any variety of things. You have to understand math to be able to survive in the world. Even men of ancient times had an understanding of math. You had to be able to barter effectively, and how could you do that if you didn’t understand mathematics. Math is a requirement in the modern world and is essential to your everyday success.

What is the greatest common factor of a polynomial? How do you know when you have found the greatest one? Explain your answers, show a demonstration and give a problem for the class to try.

To find the greatest common factor of a polynomial is rather simple. You have to break down each term of the polynomial to its basic factored out pieces. In the following equation I will show you how to find the greatest common factor of the polynomial. 10x^2y^2+ 5xy+20y. In this equation the highest common factor is 5. You must now identify the highest common variable between all elements of the polynomial. In this equation the highest common factor is just y so our new greatest common factor is 5y. When we factor this out of our polynomial we get 5y(2x^2y + x + 4x). An example for the class to try is: 15x^3y^2 -3x^2y^2+15y^2

Explain how to factor trinomials. Is there more than one way to factor trinomials? Show your answer using both words and mathematical notation. Give a problem to the class to try.

The way I factor trinomials is by taking the leading and the last term and multiplying them together. I then take the new number that those multiplied out to and figure out what two factors of that multiplied number can equal the middle number. For instance in the following trinomial 10x^2 -11x + 3. First we take and multiply the leading term, 10 by the trailing term, 3. This gives us 30. We much now factor 30 and find out which two factors of 30 when added together equal the middle term, 11. In the number 30 the factors are 1,2,3,5,6,10,15,30. Right off the bat I can see that 5 and 6 both add up to 11 and multiply to 30. Now our problem is that 5 and 6 do not add up to negative 11 so they must both be negative. So our new polynomial is, 10x^2 -5x -6x +3. To factor these you must now group and pull out the highest common factor from each problem. It is very simple! In order to factor this polynomial we have to pull a 5x out of the first grouping (10x^2-5x) (10x^2-5x) (-6x + 3) 5x(2x-1) -3(2x-1) (5x-3) (2x-1) x = 1/2 , or 3/5

Demonstrate how to factor the difference of two squares, a perfect square trinomial, and the sum and difference of two cubes. Which of these three makes the most sense to you? Give a problem for the class to factor.

To factor the difference of two squares I will use x^2 - 4 To factor this the first term is broken up in the following format: (x - ?) (x + ?) The second term is broken up into its rooted number. (? – 2) (? + 2) When combined the terms look like this (x – 2) (x + 2) A problem for the class to solve is: x^2 – 9 To factor a perfect square trinomial I will use x^2 + 8x + 16 To break the trinomial up into groupable terms you must multiply the a and the c terms and see what factors add up to the b term. In this case 4 and 4 are factors of 16 and add up to 8. x^2 + 4x +4x + 16 We then group the first two terms and the last two terms. (x^2 + 4x) ( 4x + 16) Then factor the GCF out of each grouping. x( x + 4) 4 (x+4) We are left with (x + 4) (x + 4) A problem for the class to solve is x^2 + 10x + 25 To factor the difference of two cubes I will use (x^3 – 125) (a^3 – b^3) To break up this cubes you must use the following format: (a – b)(a^2 + ab + b^2) In our example problem we will plug in our 3rd root of the first term, x and the 3rd root of the second term, 5. This end up looking like this: (x – 5) (x^2 + 5x +25) A problem for the class to solve is (x^3 – 8)

What one area from the readings in Week 3 are you most comfortable with? Why do you think that is? Using what you know about this area, create a discussion question that would trigger a discussion—that is, so there is no single correct answer to the quest

The part of the class which was most graspable to me was the part about factoring the difference of two of two cubes. I think that because there is a clear distinct formula for doing this and it doesn’t change it becomes a very easy part of the class to understand. If I were to design my own discussion question for the class I would probably ask the students to describe how someone came up with the formula for factoring the difference of two cubes. This is an abstract question that needs a bit of research, and it would spur the discussion between students.

Explain the five-steps for solving rational equations. Can any of these steps be eliminated? Can the order of these steps be changed? Would you add any steps to make it easier, or to make it easier to understand? Give a rational equation for the class to

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The First step in the process is to find the least common denominator of all the fractions in the equation. The second step is to multiply the terms with the least common denominator to eliminate the fractions. The third step is to simplify the terms. The Fourth step is to solve the equation. The fifth step is to check the solution to make sure that the solution does not make the fraction undefined. The steps cannot be reordered or eliminated. They are all necessary steps to completing the process. If steps were replaced in this sequence the data wouldn’t be consistent with the correct answer. A rational equation for the class to solve is: (x-3)/((x^2-6x+9))

Do all rational equations have a single solution? Why is that so?

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All rational equations don’t necessary have a single solution however they do have solutions that can’t fulfill the answer. Any solution where the denominator is equal to zero results in an undefined equation. This is a huge problem when solving rational equations. You must make sure that the denominator of the equation does not ever equal zero. Rational equations may have more than one solution due to rational equations having polynomials. There can be more than one solution to a single equation.

What are the two steps for simplifying radicals? Can either step be deleted? If you could add a step that might make it easier or easier to understand, what step would you add? Give a radical term for the class to simplify.

The first step to simplifying a radical is breaking it down into its perfect square numbers, for instance in the √8 This can be broken down into 2x4. Four is a perfect square number. Therefore its square root can be pulled out of the radical. In this case 2. There is however still left a two under the radical so the simplified radical looks like. 2√2. There isn’t really a step that can be eliminated in this process. I f I were to add a step or break it down for someone else to understand I would simply explain that the simplest way to break down a radical is to break it down to its LCD, and any numbers left over that have pairs of 2 can be pulled out of the radical. A radical term for the class to simplify is √98.

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Math plays a huge role in any profession. A big part of the reason that math is such a huge part of our lives is due to finances. If finances are taken care of appropriately then everyone is a happy camper. If you do not have a firm grasp on finances or on the math that takes care of finances. You can run into issues throughout your life. A specific example so far that relates to my life is the graphing. I think that slopes are an important thing to understand for expense. If you don’t have a firm grasp on expenses and the marginal cost related to everything then you can be left confused very quickly.

The acronym I use to complete this is PEMDAS. This means that equations must be completed first within the parenthesis, then the exponents, then multiplication and division followed by addition and subtraction. Typically steps can be skipped if there is not one component of the order of operation in the problem. However steps cannot be done out of order because this results in entirely different solutions then what was originally the real result of the problem. For instance in the following equation 4(5-2)^2 You can skip the Division step because there isn’t any division steps needed. For this problem though you must first work in the parenthesis by subtracting 5-2 (3). The next step is to square it, (9) and then the last step is to multiply it by 4 (36). This is how order of operations must always be done otherwise a different answer could come up.

What four steps should be used in evaluating expressions? Can these steps be skipped or rearranged? Explain your answers.

The acronym I use to complete this is PEMDAS. This means that equations must be completed first within the parenthesis, then the exponents, then multiplication and division followed by addition and subtraction. Typically steps can be skipped if there is not one component of the order of operation in the problem. However steps cannot be done out of order because this results in entirely different solutions then what was originally the real result of the problem. For instance in the following equation 4(5-2)^2 You can skip the Division step because there isn’t any division steps needed. For this problem though you must first work in the parenthesis by subtracting 5-2 (3). The next step is to square it, (9) and then the last step is to multiply it by 4 (36). This is how order of operations must always be done otherwise a different answer could come up.

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Do you always use the property of distribution when multiplying monomials and polynomials? Explain why or why not. In what situations would distribution become important?

Yes you always need to use the property of distribution to multiply two or more polynomials together. This ensures that no terms are skipped and that the final result is the entire product of the two or more polynomials. The only real exception to this rule when you have two or more monomials, where there are not any polynomials. This means that you can just multiply the monomials in any order that you really can. Distribution becomes important when you have a simple 2 degree polynomial, and then you have to multiply by another 2 or 3 degree polynomial. It is important to keep track of the order of the polynomials.

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What is the greatest common factor? How do you know when you have found the greatest one?

The greatest common factor is the term that factors out of all elements of a polynomial. The greatest common factor can be found by factoring each individual term of a polynomial. This can be a tedious process and it is often best to analyze each term of a polynomial before going through with all of that. Often times it can be easiest to identify the greatest common factor by finding out how each term is related. I will show you how to find the greatest common factor of the polynomial. 10x^2y^2+ 5xy+20y. In this equation the highest common factor is 5. You must now identify the highest common variable between all elements of the polynomial. In this equation the highest common factor is just y so our new greatest common factor is 5y. When we factor this out of our polynomial we get 5y(2x^2y + x + 4x).

Explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c. Is there more than one way to factor this? Show your answer using both words and mathematical notation.

The best method I know of to show you how to solve these trinomials is by assigning values to the a and c values. The second polynomial will hypothetically look like this. 4x^2 + 8x + 4. To factor the following trinomials you must first multiply the first term and the last terms a and c values. In the first equation there is no a value so just assume that a is 1. So you start off by multiplying a by c. Which in our equation is 4 and 4, this means that the value is 16. The next step in the process is to find out what factors of 16 add up to the middle term b. The factors of 16 are: 1,2,4,4,8,16. Right away we can see that the two factors 4 and 4 add up to the middle term in our equation b, or 8. This means that we can rewrite our polynomial in the form of ax2+bx +bx + c. You then group the two terms, factor and solve. Our polynomial should look like: 4x^2 + 4x +4x + 4

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Demonstrate how to factor the difference of two squares, a perfect square trinomial, and the sum and difference of two cubes. Which of these three makes the most sense to you?

To factor the difference of two squares I will use y^2 - 9 To factor this the first term is broken up in the following format: (y - ?) (y + ?) The second term is broken up into its rooted number. (? – 2) (? + 9) When combined the terms look like this (y – 2) (y + 9) A problem for the class to solve is: x^2 – 4 To factor a perfect square trinomial I will use y^2 + 10y + 25 To break the trinomial up into groupable terms you must multiply the a and the c terms and see what factors add up to the b term. In this case 5 and 5 are factors of 25 and add up to 10. y^2 + 5y +5y + 25 We then group the first two terms and the last two terms. (y^2 + 5y) ( 5y + 25) Then factor the GCF out of each grouping. y( y+ 5) 4 (y+5) We are left with (y + 5) (y + 5) A problem for the class to solve is x^2 + 10x + 25 To factor the difference of two cubes I will use (y^3 – 64) (a^3 – b^3) To break up this cubes you must use the following format: (a – b)(a^2 + ab + b^2) In our example problem we will plug in our 3rd root of the first term, y and the 3rd root of the second term, 5. This end up looking like this: (y – 4) (y^2 + 4y +16) A problem for the class to solve is (x^3 – 27)

What constitutes a rational expression? How would you explain this concept to someone unfamiliar with it?

A rational expression is an equation where the value of P is divided by the value of Q. The values of P and Q are actually associated with polynomials in a rational expression. In order to explain what constitutes a rational expression to someone I would explain that Q in this case cannot be equal to zero. In the following situation (x-4)/(x^2-8x+16) you have to make sure that the solution (4) is not used because it will make the denominator 0 which means that the equation will be undefined.

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Explain the five-steps for solving rational equations. Can any of these steps be eliminated? Can the order of these steps be changed? Would you add any steps to make it easier, or to make it easier to understand?

In order to solve a rational equation the first step in the process is to find the least common denominator of all the fractions in the equation. The second step is to multiply the terms with the least common denominator to eliminate the fractions. The third step is to simplify the terms. The fourth step is to solve the equation. The fifth step is to check the solution to make sure that the solution does not make the fraction undefined. These steps are not able to be reordered or eliminated. They are actually all necessary steps to completing the solving process of the equation. If steps were replaced in this sequence the data wouldn’t be consistent with the correct answer. A rational equation for the class to solve is: (x-5)/((x^2-10x+25))

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What are the two steps for simplifying radicals? Can either step be deleted? If you could add a step that might make it easier or easier to understand, what step would you add?

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The first step to simplifying a radical is breaking it down into its perfect square numbers, (by perfect square numbers I mean numbers like 4, 9, 16, 25 etc.) for instance in the √12. This can be broken down into 3x4. Four is a perfect square number. Therefore its square root can be pulled out of the radical. In this case 2. There is however still left a three under the radical so the simplified radical looks like. 2√3. There isn’t really a step that can be eliminated in this process. I f I were to add a step or break it down for someone else to understand I would simply explain that the simplest way to break down a radical is to break it down to its LCD, and any numbers left over that have pairs of 2 can be pulled out of the radical. A radical term for the class to simplify is √147.

What role do radical numbers play in your current or future profession? Provide a specific example and relate your discussion to your classroom learning this week.

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Currently radical numbers don’t play any part in my current or I believe in my future profession. I don’t run into day to day problems where I have to take the square root of any number. I think that that part of math isn’t really something that is needed for basic functioning. I don’t have a need for square roots other than to solve my math homework. I do however have needs for slopes and other mathematical like principles, but I seldom run into an occasion where I need radicals.

Imagine that a line on a graph is approximately the distance y in feet a person walks in x hours. What does the slope of this line represent? How is this graph useful? Provide another example for your colleagues to explain.

The slope in this line represents the number of feet walked in an amount of hours. This is a very useful graph for knowing how many steps you may have taken in a day. For instance if you are trying to track how many steps you take for a study, or for weight loss, or for charity this can be an excellent graph to track your progress. The x and the y labels can be easily changed to represent different things. For instance, the x and y can be changed to represent the amount of tables built in a week. This can be useful for a carpenter to know how many tables he is producing on an average basis, and whether the amount done previously is going up or staying steady. This is also useful for the average family income, and tracking that data. If I wanted my colleagues to explain something it would be the number of candles that my friend burns in a year. I can tell you it is an obscene amount, so if you can explain that it would be great. You can let x be the year or months, and y be the candles.

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