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Factoring Polynomials

Grade 10
Sep 16, 2022

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.  

Factoring Polynomials

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

greatest common factor

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.  

terms of the polynomial

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. 


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.  

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).  

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

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.  


Question 1 

Find the GCF of each term of a polynomial.  

1. 15x2 + 18   


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 


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   


x3 = x × x × x 

5x2 = 5 × x × x  

-22x = (-2) × 11 × x  

Here, GCF is x.  

x (x2 + 5x – 22)  

2. -16y6 + 28y4 – 20y3 


-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? 


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  


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.  



  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  


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