FACTORING BY GCF WORKSHEET

Problem 1 :

4x² - 3x

Problem 2 :

10y³ + 20y² - 5y

Problem 3 :

-12a - 8a²

Problem 4 :

5x² + 7

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)

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.

Problem 13 : 

If one of the factor of x² + x – 20 is (x + 5). Find the other

(a)   x – 4      (b)   x + 2      (c)   x + 4      (d)   x – 5

Problem 14 : 

If x + 2 is a factor of x³ – 2ax² + 16, then value of a is

(a)   3       (b)   1      (c)   4      (d)   2

Detailed Answer Key

1. Answer : 

4x² - 3x

Find the GCF :

4x²  =  2 ⋅ 2 ⋅ x ⋅ 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 ⋅ 5 ⋅ y ⋅ y ⋅ y

20y²  =  2 ⋅ 2 ⋅ 5 ⋅ y ⋅ y

5y  =  5 ⋅ 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 ⋅ 2 ⋅ 3 ⋅ a

8a²  =  2 ⋅ 2 ⋅ 2 ⋅ a ⋅ a

The GCF of 12a and 8a² is 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  =  5 ⋅ 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.

(13) Solution :

= x² + x – 20

By factoring the polynomial, we get

= x² - 4x + 5 x – 20

= x(x - 4) + 5(x - 4)

= (x + 5)(x - 4)

Since one of the factor is (x + 5), the other factor will be x - 4. Option a is correct.

(14)  Solution :

If x + 2 is a factor of x³ – 2ax² + 16, then value of a is

Since x + 2 is a factor of the polynomial, by equating this factor to 0, we get x = -2

Let P(x) = x³ – 2ax² + 16

P(-2) = (-2)³ – 2a(-2)² + 16

0 = -8 - 2a(4) + 16

0 = 8 - 8a

8a = 8

a = 8/8

a = 1

So, the value of a is 1.

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