FACTORING BY GCF WORKSHEET

Problems 1-4 : Factor each polynomial. 

Problem 1 : 

4x2 - 3x

Problem 2 : 

10y3 + 20y2 - 5y

Problem 3 : 

-12a - 8a2

Problem 4 : 

5x2 + 7

Problems 5-8 : Factor each expression.

Problem 5 : 

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

Problem 6 : 

-y(y2 + 5) + (y2 + 5)

Problem 7 : 

9n(n + 4) - 5(4 + n)

Problem 8 : 

-3y2(y + 2) + 4(y - 7)


Problems 9-10 : Factor each polynomial by grouping.

Problem 9 : 

12x3 - 9x2 + 20x - 15

Problem 10 : 

9a3 + 18a2 + a + 2

Problem 11 : 

Factor 3x3 - 15x2 + 10 - 2x by grouping.

Problem 12 :

Lily’s calculator is powered by solar energy. The area of the solar panel is (7x2 + x) cm2. Factor this polynomial to find possible expressions for the dimensions of the solar panel.

Detailed Answer Key

1. Answer : 

4x2 - 3x

Find the GCF :

4x2  =  2 ⋅ 2  ⋅ x

3x  =  3 ⋅ x

The GCF of 4x2 and 3x is x.

Write terms as products using the GCF as a factor.

=  4x(x) - 3(x)

Use the Distributive Property to factor out the GCF.

=  x(4x - 3)

2. Answer : 

10y3 + 20y2 - 5y

Find the GCF :

10y3  =  2 ⋅  y ⋅ y ⋅ y

20y2  =  2 ⋅ 2   y ⋅ y

5y  =   y

The GCF of 10y3, 20y2 and 5y is 5y.

Write terms as products using the GCF as a factor.

10y3 + 20y2 - 5y  =  2y2(5y) + 4y(5y) - 1(5y)

Use the Distributive Property to factor out the GCF.

=  5y(2y2 + 4y - 1)

3. Answer : 

-12a - 8a2

Both coefficients are negative. Factor out -1.

-12a - 8a2  =  -1(12a + 8a2)

Find the GCF :

12a  =  ⋅ 2  3 ⋅ a

8a2  =  ⋅ ⋅ 2  a ⋅ a

The GCF of 12a and 8ais 4a. 

Write terms as products using the GCF as a factor.

=  -1[3(4a) + 2a(4a)]

Use the Distributive Property to factor out the GCF.

=  -1[4a(3 + 2a)]

=  -4a(3 + 2a)

4. Answer :  

5x2 + 7

Find the GCF :

5x2  =   x ⋅ x

7  =  7

There are no common factors other than 1.

The polynomial cannot be factored.

5. Answer : 

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

(x - 3) is a common binomial factor.

=  7(x - 3) - 2x(x - 3)

Factor out (x - 3).

=  (x - 3)(7 - 2x)

6. Answer : 

-y(y2 + 5) + (y2 + 5)

(y2 + 5) is a common binomial factor.

=  -y(y2 + 5) + (y2 + 5)

=  -y(y2 + 5) + 1(y2 + 5)

Factor out (y2 + 5).

=  (y2 + 5)(-y + 1)

=  (y2 + 5)(1 - y)

7. Answer : 

9n(n + 4) - 5(4 + n)

Addition is always commutative. So,

4 + n  =  n + 4

Then, 

=  9n(n + 4) - 5(n + 4)

(n + 4) is a common binomial factor.

=  9n(n + 4) - 5(n + 4)

Factor out (n + 4).

=  (n + 4)(9n - 5)

8. Answer : 

-3y2(y + 2) + 4(y - 7)

There are no common factors.

The expression cannot be factored.

Factor each polynomial by grouping.

9. Answer : 

12x3 - 9x2 + 20x - 15

Group terms that have a common number or variable as a factor.

=  (12x3 - 9x2) + (20x - 15)

Factor out the GCF of each group.

=  3x2(4x - 3) + 5(4x - 3)

(4x - 3) is a common factor.

=  3x2(4x - 3) + 5(4x - 3)

Factor out (4x - 3).

=  (4x - 3)(3x2 + 5)

10. Answer : 

9a3 + 18a2 + a + 2

Group terms that have a common number or variable as a factor.

=  (9a3 + 18a2) + (a + 2)

Factor out the GCF of each group.

=  9a2(a + 2) + 1(a + 2)

(a + 2) is a common factor.

=  9a2(a + 2) + 1(a + 2)

Factor out (a + 2).

=  (a + 2)(9a2 + 1)

11. Answer :

=  3x3 - 15x2 + 10 - 2x

Group terms.

=  (3x3 - 15x2) + (10 - 2x)

Factor out the GCF of each group.

=  3x2(x - 5) + 2(5 - x)

Write (5 - x) as -1(x - 5).

=  3x2(x - 5) + 2(-1)(x - 5)

=  3x2(x - 5) - 2(x - 5)

(x - 5) is a common factor. 

=  3x2(x - 5) - 2(x - 5)

Factor out (x - 5).

=  (x - 5)(3x2 - 2)

12. Answer : 

A  =  7x2 + x

The GCF of 7x2 and x is x.

Write each term as a product using the GCF as a factor.

=  7x(x) + 1(x)

Use the Distributive Property to factor out the GCF.

=  x(7x + 1)

Possible expressions for the dimensions of the solar panel are x cm and (7x + 1) cm.

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