Section 6.1

An Introduction to Factoring Polynomials

6.1 Lecture Guide: An Introduction to Factoring Polynomials

Objective: Factor out the GCF of a polynomial.

Greatest Common Factor of a Polynomial:

The GCF of a polynomial is the common factor that contains 1. the largest possible numerical coefficient and 2. the largest possible exponent on each variable factor.

Determine the GCF of each pair of polynomials.

1. 3 2 224 36x y xyand

2. 215 25x y xyand

Determine the GCF of each pair of polynomials.

Factor out the GCF of each polynomial.

3. 3 2 224 36x y xy

Factor out the GCF of each polynomial.

4. 215 25x y xy

Factor out the GCF of each polynomial.

5. 2 2 3 315 45 30x y x y xy

Factor out the GCF of each polynomial.

6. 3 2 5 3 2a a b b a b

7. (a) Factor out the GCF of the polynomial

(b) Use your calculator to complete the table below by letting 1Y equal the original polynomial and 2Y equal thefactored form.

–3

–2

–1

0

1

2

3

x 1 1 4 1Y x x x 2Y

1 4 1 .x x x

(c) Graph

(d) What would you conclude about the original polynomial

1Y and 2Y in the standard viewing window. How do the graphs compare?

1 4 1x x x and the factored form?

7. Factor out the GCF of the polynomial 1 4 1 .x x x

Objective 2: Factor by Grouping.

Factoring a Four-Term Polynomial by Grouping Pairs of Terms

Step 1. Be sure you have factored out the GCF if it is not 1.

Step 2. Use grouping symbols to pair the terms so that each pair has a common factor other than 1.

Example: 2 2 4ax ay bx by

Factoring a Four-Term Polynomial by Grouping Pairs of Terms

Step 3. Factor the GCF out of each pair of terms.

Step 4. If there is a common binomial factor of these two groups, factor out this GCF. If there is no common binomial factor, try to use Step 2 again with a different pairing of terms. If all possible pairs fail to produce a common binomial factor, the polynomial will not factor by this method.

Factor each polynomial by the grouping method.

8. 3 8 24ax a bx b

Factor each polynomial by the grouping method.

9. 22 16 3 24x x x

Factor each polynomial by the grouping method.

10. 23 9 2 6x x x

Factor each polynomial by the grouping method.

11. 26 8 15 20x x x

Objective 3: Use the zeros of a polynomial and the x-intercepts of the graph of to factor the polynomial.

The relationship among the factors of a polynomial, the zeros of a polynomial function, and the x-intercepts of a graph of a polynomial function is an important one.

If is an input value for which the output equals 0, then c is called a zero of the function. x c

y P x P x

P c

For a real constant c and a real polynomial , the following statements are equivalent.

Graphically Numerically Algebraically(c, 0) is an x-intercept of the graph of .

, that is, c is a zero of .

is a factor of .

P x y P x 0P c

P xx c

P x

Equivalent Statements about Linear Factors of a Polynomial

12. Consider the polynomial 1 5 .P x x x Use thefactored form of P x to evaluate each expression.

(a) 5P (b) 3P

(c) 1P (d) 1P

1 5P x x x 13. Use the table and graph provided for to answer each question.

−5 0

−4 −5

−3 −8

−2 −9

−1 −8

0 −5

1 0

x P x

-7 -5 -3 -1 1 3

-10

-5

0

5

10

y

x

(a) List the factors of .P x

(b) List the zeros of .P x

(c) List the x-intercepts of the graph.

14. Consider the polynomial 2 2 3.P x x x

-5 5

-5

5

y

x

P x

−4 5

−3 0

−2 −3

−1 −4

0 −3

1 0

2 5

x P x (a) List the zeros of .P x

(b) List the x-intercepts of the graph.

(c) Use parts (a) and (b) to determine the factored form of the polynomial 2 2 3.P x x x

Use the factored form of each polynomial to list the zeros and the x-intercepts of the graph of

15.

Zeros:

x-intercepts:

3 4P x x x

.y P x

Use the factored form of each polynomial to list the zeros and the x-intercepts of the graph of

16.

Zeros:

x-intercepts:

5 4 2P x x x x

.y P x

Polynomial Factored Form

Zeros of x-intercepts of the graph of

17.

18. 8 and 10

19.

20. –7, 0, and 9

y P x

2 3 54x x 6 9x x

1,0 , 6,0

y P x

2 18 80x x

2 5 6x x

3 22 63x x x

Complete the following table for each polynomial.

Use the given graph for to factor this polynomial.

21. 2 2 8P x x x

-6 -4 -2 0 2 4

-10

-5

0

5

10

y

x

2 2 8P x x x

Use the given table for to factorthis polynomial.

22.

0 01 52 03 −94 −165 −156 0

P xx

3 28 12P x x x x

23. Use a graphing calculator to create a graph for Then use the zeros to factor the polynomial.

Sketch:

Zeros:

-10 -5 0 5 10

-10

0

10

20

30

40

y

x Factored form:

3 23 10 24.P x x x x

24. Use a graphing calculator or a spreadsheet to complete The table for Then use the zeros to factor the polynomial.

4 210 9.P x x x

Zeros:

Factored form:

−3−2−10123

x P x

Factoring a Polynomial with Two Variables Use the factored form of a polynomial in one variable to factor each polynomial in two variables.

25. Given: 2 14 24 2 12x x x x

Factor: 2 214 24x xy y

Factoring a Polynomial with Two Variables Use the factored form of a polynomial in one variable to factor each polynomial in two variables.

26. Given:

Factor:

2 5 36 4 9x x x x

2 25 36x xy y