Factoring Polynomials

Factoring Polynomials

Please have the notes for chapter 6 on hand when you do these lessons.

Factoring polynomials

• Last chapter we were combing terms in order to have polynomials, now we will go in reverse and factor to get to the terms used in its make up.

• Review how to factor integers.

• 30 can be factored to its prime values of 2 x 3 x 5.

• 48 can be factored to its prime values of 2 x 2 x 2 x 2 x 3

Multiple Choice

Use a factor tree to find the prime factorization of 64

• 4x4x4

• 4x4x2x2

• 2x2x2x2x2x2

• 4x4x2x1

Multiple Choice

Use a factor tree to find the prime factorization of 32

• 2x2x2x4

• 2x2x2x2x2

• 4x4x2

• 2x2x2x2x1

Multiple Choice

Use a factor tree to find the prime factorization of 60

• 5x3x2x2

• 7x6x5

• 5x3x3x2

• 7x5x4

Multiple Choice

2 x 2 x 2 is the prime factorization of

• 6
• 12
• 22
• 8

Multiple Choice

Is 5 prime or composite? 

• Prime
• Composite

Greatest Common Factors (GCF)

• Finding the greatest common factor between values helps reduce a whole number.

• When finding the common factor of a variable, you look at the exponent values. Remember the exponent rules we just covered in chapter 5.

• Ex. Find the greatest common factor for

   $3x^2\ +\ 12\ x$  

• The GCF for the whole number is 3

• Factoring that out leave  $3\left(x^2\ +\ 4x\right)$  

• Now factor out the variable: greatest common factor is x

• Completely factored: 3x(x + 4)

Multiple Choice

What is the GCF of the terms below?

24x³ + 36x²

• 8x

• 6x²

• 12x²

• 4x

• 12x³

Multiple Choice

Factor using GCF or monomial factoring:

x² - 5x

• 1(x - 5)

• x(x - 5x)

• x(x - 5)

• x²(1 - 5)

Multiple Choice

Factor out the GCF:
10x³ - 15x

• x(10x² - 15)
• 5x(2x² - 3)
• 5(2x³ - 3x)
• 5x²(10x - 3)

Multiple Choice

Factor using GCF or monomial factoring:

2n²-8n³

• 2n²(n-4n²)

• 2n²(1-4n)

• 2n³(10-8n²)

• 3n(1-4n)

Multiple Choice

Which is a common factor of 12x and 27?

• x

• 12

• 3

• 9

Factoring by Grouping

• When factoring by grouping you will always have four terms.

•  $3x^2\ +\ 3x\ +\ 5xy\ +\ 5y$  

• Group the common terms in pairs

•  $\left(3x^2\ +\ 3x\right)\ +\left(5xy\ +\ 5y\right)$  

• Now find the GCF in each set of terms

• 3x(x+1) + 5y(x + 1)

• Do you see a common term after they were factored?

• Factor the greatest term from the factored binomials

• (x + 1)

• The terms left make the factoring complete.

• (x + 1) (3x + 5y)

Multiple Choice

Factor the Polynomial

x³ - x² + 2x - 2

• (x² + 2) (x - 1)

• (x² - 2) (x + 1)

• (x² + 1) (x - 2)

• (x² - 1) (x + 2)

Multiple Choice

Factor the Polynomial

3x³ - 4x² + 9x - 12

• (x² + 3) (3x - 4)

• (x² - 3) (3x + 4)

• (3x² + 3) (x - 4)

• (3x² - 3) (x + 4)

Multiple Choice

Factor the Polynomial

2x³ + 5x² + 6x + 15

• (x² + 3) (2x + 5)

• (x² - 3) (2x - 5)

• (2x² - 3) (x - 5)

• (2x² + 3) (x + 5)

Multiple Choice

Factor the Polynomial

12x³ - 9x² + 4x - 3

• (3x² + 1) (4x - 3)

• (3x² - 1) (4x + 3)

• (4x² + 1) (3x - 3)

• (4x² - 1) (3x + 3)

Factoring Trinomials

 $x^2\ +\ bx\ +\ c$  

• The signs for this can be either positive or negative.  Pay close attention to the signs.

• There are two methods of factoring these: grouping or finding common terms. 

• The next example will be factored both ways.

• 
  

 $x^2\ -\ 7x\ +\ 10$  

• Set up two sets of  ()().

• Now you must find multiples of 10 that add up to - 7.  Multiples of 10 could be -2 x -5.  That will give 10 and when they are added it will give -7.  

• ( x - 5) (x - 2)

• Check by multiplying to see if the answer at the top is the result.

Multiple Choice

Factor the perfect square trinomial

x² - 18x + 81

• (x - 9)(x + 9)

• (x - 3)(x + 9)

• (x + 9)(x + 9)

• (x - 9)(x - 9)

Multiple Choice

Factor the expression

 $x^2-x-6$  

• (x+2)(x-3)

• (x-2)(x+3)

• (x-2)(x-3)

• x(x-3)

Multiple Select

What are the binomial factors of  $x^2+7x-18$  

(Pick 2 answers)

•  $\left(x-9\right)$  

•  $\left(x+9\right)$  

•  $\left(x-2\right)$  

•  $\left(x+2\right)$  

Multiple Choice

What is the factored form of the expression?  $x^2+x-6$  

•  $\left(x-3\right)\left(x+2\right)$  

•  $\left(x+1\right)\left(x-6\right)$  

•  $\left(x-2\right)\left(x+3\right)$  

•  $\left(x-2\right)\left(x-3\right)$  

Multiple Choice

Which is a factor of  $x^2-11x+24?$  

•  $\left(x+3\right)$  

•  $\left(x-3\right)$  

•  $\left(x+4\right)$  

•  $\left(x-4\right)$