Factoring Completly

Factoring Completly

Factoring Completely

1) Is there a GCF?

2) How many terms are left to be factored?

3) If it is a trinomial, can it be factored using the X method?

4) If it is a binomial, is it a difference of squares? A sum of cubes? A difference of cubes?

Factoring Polynomials

1) Is there a GCF? Yes, 3

3(x^2 - 4x + 3)

2) How many terms need to be factored? 3

3) Can we use the X method? 2 numbers that multiply to ac (3) and add to b (-4)? Yes, -3 and -1

3(x-3)(x-1)

Factoring Binomials

1) Is there a GCF? Yes, 2.

2(64x^3 + 27)

2) How many terms are left to be factored? 2

Is it a difference of squares? No

Is it a sum of cubes? Yes: Use the sum of cubes structure

(a+b)(a^2 - ab + b^2)

a = 4x, b = 3

2(4x + 3)(16x^2 - 12x + 9)

Polynomials with 4 terms

1) Is there a GCF? No

2) How many terms need to be factored? 4 - Double Grouping

(3x^3 - 12x) + (7x^2 - 28)

3x(x^2 - 4) + 7(x^2 - 4)

(3x + 7)(x^2 - 4)

3) Are either of the binomials difference of squares, sum of cubes, or difference of cubes? YES - difference of squares

(3x + 7)(x + 2)(x - 2)

Open Ended

Describe the first step in factoring a polynomial. Why is this the first step? Explain.

Fill in the Blanks

Determine the GCF for the polynomial:

 $2x^3-7x^2+3x$  

Fill in the Blanks

Determine the GCF for the polynomial:

 $33x^4-22$  

Fill in the Blanks

Determine the GCF for the polynomial:

 $24x^5-56x^3+16x$  

Multiple Select

Which polynomials are prime? Check all that apply.

• $2x^2+7x+1$

• $5x^2-10x+5$

• $3x^2+8$

• $4x^2-25$

• $x^2+36$

Multiple Choice

Factor the polynomial completely. Which is the completely factored polynomial?

 $4x^3-12x^2+7x-21$  

•  $4x^2\left(x+4\right)$  

•  $3x^2\left(x-3\right)$  

•  $\left(4x^2-7\right)\left(x+3\right)$  

•  $\left(4x^2+7\right)\left(x-3\right)$  

Multiple Choice

Which diagram shows the X method being used correctly to factor the trinomial?

 $2x^2-3x+1$  

Multiple Choice

Factor the polynomial completely:

 $6x^2-8x+2$  

•  $\left(3x-1\right)\left(x-1\right)$  

•  $\left(3x+1\right)\left(x+1\right)$  

•  $2\left(3x-1\right)\left(x-1\right)$  

•  $2\left(3x+1\right)\left(x+1\right)$  

Multiple Choice

Completely factor the polynomial:

 $8x^2-24x+20x-60$  

•  $8x\left(x+3\right)\left(x-5\right)$  

•  $\left(4x-3\right)\left(2x-5\right)$  

•  $2\left(2x-3\right)\left(2x+5\right)$  

•  $4\left(x-3\right)\left(2x+5\right)$  

Multiple Choice

Which product of prime polynomials is equivalent to

 $30x^3-5x^2-60x$  ?

•  $5\left(2x^2-3\right)\left(3x+4\right)$  

•  $x\left(10x+3\right)\left(3x-4\right)$  

•  $5x\left(2x-3\right)\left(3x+4\right)$  

•  $5x\left(2x+3\right)\left(3x-4\right)$  

Multiple Select

Which are the prime factors of the polynomial

 $24x^4-3x$  ? Check all that apply. 

•  $3x$  

•  $\left(4x^2+1\right)$  

•  $\left(2x-1\right)$  

•  $\left(x-1\right)$  

•  $\left(4x^2+2x+1\right)$  

Open Ended

On a test, Leo is asked to completely factor the polynomial

 $3x^3-3x+5x^2-5$  . He uses double grouping to get  $\left(x^2-1\right)\left(3x+5\right)$  . Has he factored the polynomial completely? Explain