OBJECTIVES 5.1 Introduction to Factoring Slide 1Copyright 2011, 2007, 2003, 1999 Pearson Education,...

OBJECTIVES

5.1 Introduction to Factoring

Slide 1Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

a Find the greatest common factor, the GCF, of monomials.

b Factor polynomials when the terms have a common factor, factoring out the greatest common factor.

c Factor certain expressions with four terms using factoring by grouping.


Multiply


3 (a + 1) = 3a + 3

Factor • Factor = Product

Factor • Factor = Polynomial


Factoring


To factor a polynomial is to find an equivalent expression that is a product.Since we know:

Factor • Factor = Polynomial 3 (a + 1) = 3a + 3 We have the factors of 3a + 3.


Factoring


So, 3 is a factor of 3a +3 and (a + 1) is a factor of 3a + 3

polynomial = factor • factor 3a + 3 = 3 (a +1) An equivalent expression of this type is called a factorization of the polynomial.

20 and 30 have several factors in common.

The largest of these common factors is the greatest common factor, GCF.

One way to find the GCF is by making a list of the factors of each number.



(continued)


The factors of 20: 1, 2, 4, 5, 10, and 20

The factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30

Common numbers: 1, 2, 5, and 10.

The GCF is 10.



(continued)


Another way to find the GCF.

Find the prime factorization of each number. Then draw lines between common factors.




EXAMPLESolution

20 = 2 • 2 • 5

30 = 2 • 3 • 5

Draw lines between

the common factors.

The GCF is 10.



A Find the GCF of 20 and 30.


EXAMPLESolution : Write the prime factorization of each number.

420 = 2 · 2 · 3 · 5 · 7

924 = 2 · 2 · 3 · 7 · 11



B Find the GCF of 420 and 924.


So the GCF is 2 · 2 · 3 · 7 = 84

To factor a polynomial is to express it as a product. factor • factor

A factorization of a polynomial is an expression that names that polynomial as a product.


Factor; Factorization


EXAMPLE

x · x · x

x · x · x · x

SolutionPrime factorization of each coefficient:30x3 = 2 · 3 · 5 ·

–48x4 = –1 · 2 · 2 · 2 · 2 · 3 ·

The GCF of the coefficients is 6.



C Find the GCF of 30x3, –48x4, 54x5, and 12x2.


The GCF of these monomials is x3, because 3 is the smallest exponent of x.The GCF is 6x3.

x3

x4

1. Find the prime factorization of the coefficients, including –1 as a factor if any coefficient is negative.

2. Determine any common prime factors of the coefficients. For each one that occurs, include it as a factor of the GCF. If none occurs, use 1 as a factor.


To find the GCF of Two or more Monomials

(continued)


3. If any variable appears as a factor of all the monomials, include it as a factor, using the smallest exponent of the variable. If novariable occurs in all the monomials, use 1 as a factor.

4. The GCF is the product of the results of steps (2) and (3).


To find the GCF of Two or more Monomials

(continued)


EXAMPLE

Solution

9a 21 = (3 · 3 · a) - (3 · 7 ) Factoring each term = 3(3a – 7) Factoring out the GCF

Check: 3(3a 7) = 9a 21



D Factor: 9a 21.


EXAMPLE

Solution

28x6 + 32x3 = (4 · 7 · ) + (4 · 8 · ) Factoring each term = 4x3 (7x3 +8) Factoring out the GCF



E Factor: 28x6 + 32x3.


x6 x3

Factoring When Terms Have a Common FactorTo factor a polynomial with two or more terms of the form ab + ac, we use the distributive law with the sides of the equation switched: ab + ac = a(b + c).

Multiply Factor 4x(x2 + 3x – 4) 4x3 + 12x2 – 16x 4x·x2 + 4x·3x – 4x·4 4x·x2 + 4x·3x – 4x·4 4x3 + 12x2 – 16x 4x(x2 + 3x – 4)




EXAMPLE

What is the largest common constant factor? What is the largest common variable factor? Factor out 3x3 12x5 21x4 + 24x3

3x3(4x2 7x + 8)



F Factor: 12x5 21x4 + 24x3


3x3

EXAMPLE

Solution 9a3b4 + 18a2b3

9 a2 b3 (ab + 2)

The largest common factor is 9a2b3.



G Factor: 9a3b4 + 18a2b3


EXAMPLE

Solution 4xy + 8xw 12x = 4x(y 2w + 3)



H Factor: 4xy + 8xw 12x


STUDY TIPTips for Factoring

Before doing any other kind of factoring, first try to factor out the GCF.

Always check the result of factoring by multiplying.


Factoring by Grouping

Sometimes algebraic expressions contain a common factor with two or more terms.




EXAMPLE

Solution: The binomial (x + 2) is a factor of both x2(x + 2) and 3(x + 2). Thus, x + 2 is a common factor.

x2(x + 2) + 3(x + 2) (x2 + 3) (x + 2)

The factorization is (x2 + 3)(x + 2).



I Factor x2(x + 2) + 3(x + 2).


If a polynomial can be split into groups of terms and the groups share a common factor, then the original polynomial can be factored. This method, known as factoring by grouping, can be tried on any polynomial with four or more terms.




EXAMPLEa) 3x3 + 9x2 + x + 3 = (3x3 + 9x2) + (x + 3)

Don’t forget to include the 1.



J Factor by grouping.

(continued)


= 3x2 + 1 (x + 3)(x + 3)

EXAMPLE

b) 9x4 + 6x 27x3 18

(9x4 + 6x) + (27x3 18)

3x(3x3 + 2) + (9)(3x3 + 2)

(3x – 9)(3x3 + 2)



J Factor by grouping


OBJECTIVES 5.1 Introduction to Factoring Slide 1Copyright 2011, 2007, 2003, 1999 Pearson Education,...

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