This document discusses several topics relating to polynomials including: - Standard form of polynomials and arranging terms in descending order by exponent - Leading coefficients and degrees of polynomials - Factoring polynomials using greatest common factors, difference of squares, factoring trinomials, and factoring cubes - Dividing polynomials using long division and synthetic division - Graphing polynomial functions using calculators to find zeros and analyze characteristics like end behavior, maximums/minimums, and changes in direction - Solving polynomial equations using synthetic division and possible real roots based on factors of coefficients

1. Polynomials Topic 12.2 Factor and Divide Polynomials Standard Form of Polynomials – arrange the terms by the polynomial by the exponents of the variable in descending order. Example: Which term would be first? Which term would be last? Arrange the terms in descending order. Leading Coefficient- the number in front of the variable with the highest degree. What is the leading coefficient of the expression above?

3. Solutions

5. Long Division How many times does x go into x^2? Use regular division techniques to Solve this problem. Try it.

7. Same problem using synthetic division. Set x+5=0 and solve. Your value is called r. So in this problem r = -5 Now take the polynomial and make sure it is in descending order. Take the coefficients only and create a table. -5 / 1 13 40 Now bring the 1 down and start multiplying and dividing. I will demonstrate on the next blank side.

9. Graphing Polynomial Functions Use your graphing calculator to find the roots or zeros of the functions. Notice that the degree is 3 so this is a cubic function. What characteristics do you notice about this function? What happens At the far right and far left of the function? This is called the end behavior. This function looks like the following picture.

10. Now use the calculate function of the calculator To find the roots. (Where if crosses the x-axis.) One answer is -1. What are the other two answers? Use your calculator to find them. How many maximums and minimums does this Function have? How many times does it change Directions?

11. Solving polynomial Equations using Synthetic Division The leading coefficient is 1 and the constant term is 6. What are the factors of 1 and 6? There are only two factors of 1 and they are +1 and -1. The Factors of t are positive and negative 1,2,3,and 6. Now the possible real roots will be found by taking the factors of 6 and putting Them over all the factors of 1. (Often referred to p/q ) Those are the possible number that could be real roots. Plus or minus 1,2,3, And 6. Now use synthetic division to find if we have real roots.

12. I will try to use the pencil to do the problem. Sorry if it is messy, but it is hard To type the problem.