Section 3.1 Homework Questions?

Section

Concepts

3.1 Greatest Common Factor and Factoring by Grouping

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1. Identifying the Greatest Common Factor2. Factoring out the Greatest Common Factor3. Factoring out a Negative Factor4. Factoring out a Binomial Factor5. Factoring by Grouping

Section 3.1 Greatest Common Factor and Factoring by Grouping

1. Identifying the Greatest Common Factor

(continued)

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To factor an integer means to write the integer as a product of two or more integers. To factor a polynomial means to express the polynomial as a product of two or morepolynomials.

Section 3.1 Greatest Common Factor and Factoring by Grouping

1. Identifying the Greatest Common Factor

(continued)

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We begin our study of factoring by factoring integers. The number 20, for example, can be factored as

The productconsists only of prime numbers

and is called the prime factorization.

Section 3.1 Greatest Common Factor and Factoring by Grouping

1. Identifying the Greatest Common Factor

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The greatest common factor (denoted GCF) of two or more integers is the greatest factor common to each integer.

Example 1 Identifying the GCF

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Find the greatest common factor.

a. 24 and 36 b. 40 and 60 c. 12, 24, and 30

TIP:

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Notice that the expressions andshare a common numerical factor of 5, and common variable factors of a, and b. The GCF is the product of the largest common numerical factor (5) and the common variable factors, where each variable is raised to the lowest power to which it occurs in all the original expressions.

The GCF of and is 415a b

415a b 3 225a b

3 225a b 35a b

Example 2 Identifying the Greatest Common Factor

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Find the GCF among each group of terms.

a. b. c.

ExampleSolution:

3 Identifying the Greatest Common Factor

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Find the GCF among each group of terms.

a. , , b. , , 618x 727x 245x4 2 316r s t 5 232rs t 3 2 548r s t

Example 3 Finding the Greatest Common Binomial Factor

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Find the greatest common factor between the terms:

ExampleSolution:

3 Finding the Greatest Common Binomial Factor

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The only common factor is the binomial

The GCF is

The Greatest Common Factors in Examples 1 and 2 were monomials. The GCF In Example 3 was a binomial. Common factors which are binomials occur frequently when we “factor by grouping” a skill we will learn later in this presentation.

Section 3.1 Greatest Common Factor and Factoring by Grouping

2. Factoring out the Greatest Common Factor

(continued)

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The process of factoring a polynomial is the reverse process of multiplying polynomials. To factor out the greatest common factor means to express the original polynomial as the product of the GCF and some new factor (polynomial).

Section 3.1 Greatest Common Factor and Factoring by Grouping

2. Factoring out the Greatest Common Factor

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Multiply/Distribute

Factor out the greatest common factor

2 3 25 3 1 5 15 5y y y y y y

3 2 25 15 5 5 3 1y y y y y y

PROCEDURE Factoring out the Greatest Common Factor

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Step 1 Identify the GCF of all terms of the polynomial.Step 2 Write the original polynomial as the product of the GCF and another factor.

Note: To check the factorization, multiply the polynomials to remove parentheses.

Example 4 Factoring out the Greatest Common Factor

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Factor out the GCF.

a. b.

Example 5 Factoring out the Greatest Common Factor

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Factor out the GCF.

a. b.3 216 32 48x x x 3 212 30 42x y x y xy

Section 3.1 Greatest Common Factor and Factoring by Grouping

2. Factoring out the Greatest Common Factor

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The greatest common factor of the polynomialIf we factor out the GCF, we have 1A polynomial whose only factors are itself and 1 is calleda prime polynomial. A prime polynomial cannot be factored further.

Avoiding Mistakes

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In Example 5(a), the GCF is 16x, and the factorization of is . If the Greatest Common Factor has been factored out the second factor must be prime ( have no other factors except one).If in Example 5(a) if a GCF of 8x were used the factorization would have been . The second factor is not prime as it contains a common factor of 2. When factoring out the Greatest Common Factor remember to check to make sure the second factor is prime to ensure you have removed the GCF.

3 216 32 48x x x 216 2 3x x x

28 2 4 6x x x 22 4 6x x

Section 3.1 Greatest Common Factor and Factoring by Grouping

3. Factoring out a Negative Factor

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Usually it is advantageous to factor out the opposite of the GCF when the leading coefficient of the polynomial is negative.Notice that this changes the signs of the remaining terms inside the parentheses.

Example 6 Factoring out a Negative Factor

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Factor out from the polynomial

Example 7 Factoring out a Negative Factor

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Factor out the quantity from the polynomial

Section 3.1 Greatest Common Factor and Factoring by Grouping

4. Factoring out a Binomial Factor

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The same process which was used to factor out a common factor which was a monomial can also be used to factor out a common factor that consists of more than one term.

Example 8 Factoring out a Binomial Factor

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Factor out the GCF:

Section 3.1 Greatest Common Factor and Factoring by Grouping

5. Factoring by Grouping

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Given a four-term polynomial, we will factor it as a product of two binomials. The process is called factoring by grouping.

PROCEDURE Factoring by Grouping

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To factor a four-term polynomial by grouping:

Step 1 Identify and factor out the GCF from all four terms. (Should one exist)Step 2 Factor out the GCF from the first pair of terms. Factor out the GCF from the second pair of terms. (Sometimes it is necessary to factor out the opposite of the GCF.)Step 3 If the two terms share a common binomial factor, factor out the binomial factor.

Example 9 Factoring by Grouping

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Factor by grouping:

ExampleSolution:

10 Factoring by Grouping

(continued)

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Factor by grouping 3 23 4 15 20x x x

ExampleSolution:

11 Factoring by Grouping

(continued)

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Factor by grouping 3 24 20 3 15x x xy y

TIP:

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One frequently asked question when factoring iswhether the order can be switched between the factors. The answer is yes. Because multiplication iscommutative, the order in which the factors are written does not matter.

Example 12 Factoring by Grouping

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Factor by grouping.

Section 3.1 Greatest Common Factor and Factoring by GroupingYou Try

3 56x ya.

b. 3 215 12x y

4 221x y,Identify the GCF

Factor out the GCF

Section Greatest Common Factor and Factoring by GroupingYou Try

212 6 4x x a. b. 3 215 45 60x y x y xy

3.1

Factor

Section Greatest Common Factor and Factoring by GroupingYou Try

a. b.

3.1

Factor by Grouping3 212 18 14 21x x x 3 25 2 10x x x