Factoring

Chapter 5

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5.1 – Factoring a Monomial from a Polynomial

5.2 – Factoring by Grouping

5.3 – Factoring Trinomials of the Form ax2 + bx + c, a = 1

5.4 – Factoring Trinomials of the Form ax2 + bx + c, a ≠ 1

5.5 – Special Factoring Formulas and a General Review of Factoring

5.6 – Solving Quadratic Equations Using Factoring

5.7 – Applications of Quadratic Equations

Chapter Sections

§ 5.1

Factoring a Monomial from

a Polynomial

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Factors

To factor an expression means to write the expression as a product of its factors.

If a · b = c, then a and b are of c. a·b

factorsfactors

Recall that the greatest common factor (GCF) of two or more numbers is the greatest number that will divide (without remainder) into all the numbers.

Example: The GCF of 27 and 45 is 9.

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Factors

A prime number is an integer greater than 1 that has exactly two factors, 1 and itself.

A composite number is a positive integer that is not prime.

Prime factorization is used to write a number as a product of its primes.

24 = 2 · 2 · 2 · 3

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Determining the GCF1. Write each number as a product of

prime factors.2. Determine the prime factors common to

all the numbers.3. Multiply the common factors found in

step 2. The product of these factors is the GCF.

Example: Determine the GCF of 24 and 30.

24 = 2 · 2 · 2 · 3 30 = 2 · 3 · 5

A factor of 2 and a factor of 3 are common to both, therefore 2 · 3 = 6 is the GCF.

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Determining the GCF

To determine the GCF of two or more terms, take each factor the largest number of times it appears in all of the terms.

Example:

a.) The GCF of 6p, 4p2 and 8p3 is 2p.

2·3·p 22··22··pp·p·p 2·2 ·2·p·p·p

b.) The GCF of 4x2y2, 3xy4, and 2xy2 is xy2.

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Factoring Monomials from Polynomials

1. Determine the GCF of all the terms in the polynomial.

2. Write each term as the product of the GCF and its other factors.

3. Use the distributive property to factor out the GCF.

(To check, multiply the factors using the distributive property. )

Example: 24x + 16x3 (GCF is 8x)

= 8·3·x + 8·2·x·x2 = 8x (3 + 2x2)