Factoring Polynomials

Key Concepts

• Factor a Polynomial Model
• Find the Greatest Common Factor
• Factor out the Greatest Common Factor

Factoring Polynomials

1. Find the Greatest Common Factor 

What is the Greatest Common Factor (GCF) of the terms of 12x⁵ + 8x⁴ – 6x³? 

Step 1. Write the prime factorization of the coefficient for each term to determine if there is a greater common factor other than 1.  

Step 2. Determine the greatest common factor for the variables of each term.  

The greatest common factor of the terms 12x⁵ + 8x⁴ – 6x³ is 2x³.  

2. Factor out the Greatest Common Factor 

Why is it helpful to factor out the GCF from a polynomial? 

Consider the polynomial -12x³ + 18x+2 – 27x.  

Step 1. Find the GCF of the terms of the polynomial, if there is one. Because the first term is negative, it is helpful to factor out -1.  

The greatest common factor is -3x.  

Step 2. Factor the GCF out of each term of the polynomial.  

-3x (4x² – 6x + 9) 

Factoring out the greatest common factor results in a polynomial with smaller coefficients and/or smaller exponents of the variable(s). This makes it easier to analyze the polynomial or factor it further. 

Application 

Alani is in charge of marketing for a travel company. She is designing a brochure that will have 6 photos. The photos can be arranged on the page in a number of ways.  

1. What is the total area of the photos? 

First, find the area of each type of photo.  

Area = Area of square photos + area of narrower photos 

= 2(x²) + 4(1x)  

= 2x² + 4x 

The total area of the photos is 2x² + 4x square in.  

2. Find a rectangular arrangement for the photos. What factored expression represents the area of the arrangement? 

Try placing the photos in one row.  

The factored form that represents the area of the arrangement is x(2x + 4).  

3. Factor out the GCF from the polynomial. What does the GCF represent in this situation? 

The GCF of 2x² and 4x is 2x. So, you can rewrite the expression as 2x(x + 2).  

The GCF represents the height of one possible arrangement of the photos.  

4. Which of these two arrangements is a practical use of the space on a page of the brochure? 

The arrangement based on the GCF is more practical because the arrangement with the photos in one line will likely be too wide for a page.  

Questions  

Question 1 

Find the GCF of each term of a polynomial.  

1. 15x² + 18   

Solution: 

GCF of the coefficients:  

15 = 3×5  

18 = 2×3×3  

Here GCF is 3.  

GCF of the variables: The only common factor between x² and x⁰ is x⁰, i.e., 1. 

So, GCF is 3.  

2. -18y⁴ + 6y³ + 24y² 

Solution: 

GCF of the coefficients:  

-18 = (-1) ×2×3×3 

6 = 2×3  

24 = 2×2×2×3  

Here GCF is 2 × 3 = 6  

GCF of the variables: 

Y⁴ = y×y×y×y 

Y³ = y×y×y 

Y² = y×y  

Here GCF is y × y i.e., y².  

So, GCF is 6y².   

Question 2 

Factor out the GCF from each polynomial.  

1. x³ + 5x² – 22x   

Solution: 

x³ = x × x × x 

5x² = 5 × x × x  

-22x = (-2) × 11 × x  

Here, GCF is x.  

x (x² + 5x – 22)  

2. -16y⁶ + 28y⁴ – 20y³ 

Solution: 

-16y⁶ = (-1) × 2 × 2 × 2 × 2 × y × y × y × y × y × y  

28y⁴ = (-1) × 2 × 2 × (-7) × y × y × y × y  

-20y³ = (-1) × 2 × 2 × 5 × y × y × y  

Here, GCF is (-1) × 2 × 2 × y × y × y i.e. -4y³.  

-4y³ (4y³ – 7y + 5)  

Question 3 

In the last example mentioned in the previous section, suppose the dimensions of the narrower photos were increased to 2 in. by x in. What expression would represent the new arrangement based on the GCF? 

Solution: 

Area of square photos = 2(x²)  

Area of narrower photos = 4(2x) = 8x  

Total area of the photos = 2x² + 8x  

2x² = 2 × x × x 

8x = 2 × 2 × 2 × x  

So, GCF is 2 × x, i.e., 2x.  

2x² + 8x = 2x (x + 4)  

Area of the new arrangement based on the GCF = 2x (x + 4) square in.  

Key Concepts Covered  

Words 

Determine if a polynomial can be factored. If the polynomial can be factored, find the greatest common factor of the terms and factor it out.  

Exercise

1. How is factoring a polynomial similar to factoring integers?
2. Why does the GCF of the variables of a polynomial have the least exponent of any variable term in the polynomial?
3. What is the greatest common factor of two polynomials that do not appear to have any common factors?
4. Andrew factored 3x^2y – 6xy² + 3xy as 3xy(x – 2y). Describe and correct his error.
5. What term and 12x^2y have a GCF of 4xy? Write an expression that shows the monomial factored out of the polynomial.
   • Find the GCF of the terms of the following polynomials:
6. 8x⁶ + 32x³
7. 15x + 27
8. 7x⁴ – x³
9. 6ab² + 8ab – b
10. 86ab² + 64b – 34a