Introduction to factoring polynomials

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Slide 1

INTRODUCTION TO FACTORING POLYNOMIALS

MSJC ~ San Jacinto Campus

Math Center Workshop Series

Janice Levasseur

Slide 2

Definitions

Recall: Factors of a number are the numbers that divide the original number evenly.

Writing a number as a product of factors is called a factorization of the number.

The prime factorization of a number is the factorization of that number written as a product of prime numbers.

Common factors are factors that two or more numbers have in common.

The Greatest Common Factor (GCF) is the largest common factor.

Slide 3

Ex: Find the GCF(24, 40).

Prime factor each number:

24

40

2

12

2

6

2

3

 24 = 2*2*2*3 = 23*3

2

20

2

10

2

5

 40 = 2*2*2*5 = 23*5

 GCF(24,40)

= 23 = 8

Slide 4

The Greatest Common Factor of terms of a polynomial is the largest factor that the original terms share

Ex: What is the GCF(7x2, 3x)

7x2 = 7 * x * x

3x = 3 * x

The terms share a factor of x

 GCF(7x2, 3x) = x

Slide 5

Ex: Find the GCF(6a5,3a3,2a2)

6a5 = 2*3*a*a*a*a*a

3a3 = 3*a*a*a

2a2 = 2*a*a

The terms share two factors of a

 GCF(6a5,3a3,2a2)= a2

Note: The exponent of the variable in the GCF is the smallest exponent of that variable the terms

Slide 6

Definitions

To factor an expression means to write an equivalent expression that is a product

To factor a polynomial means to write the polynomial as a product of other polynomials

A factor that cannot be factored further is said to be a prime factor (prime polynomial)

A polynomial is factored completely if it is written as a product of prime polynomials

Slide 7

To factor a polynomial completely, ask

Do the terms have a common factor (GCF)?

Does the polynomial have four terms?

Is the polynomial a special one?

Is the polynomial a difference of squares?

a2 – b2

Is the polynomial a sum/difference of cubes?

a3 + b3 or a3 – b3

Is the trinomial a perfect-square trinomial?

a2 + 2ab + b2 or a2 – 2ab + b2

Is the trinomial a product of two binomials?

Factored completely?

Slide 8

Ex: Factor 7x2 + 3x