# HOW TO FACTOR BINOMIALS

A factor is a number that is multiplied by another number to get a product. To factor is to write a number or an algebraic expression as a product.

The following steps will be useful to factor a binomial.

Step 1 :

Find the largest common divisor for both the terms in the given binomial.

Step 2 :

Divide each term of the binomial by the largest common divisor.

Step 3:

Write the quotients inside the parentheses.

Step 4 :

Write the largest common divisor and the parentheses together using multiplication.

Factor each of the following.

Example 1 :

8m + 6

Solution :

= 8m + 6

The largest common divisor for 8m and 6 is 2.

Divide 8m and 6 by 2.

Write the quotients 4m and 3 inside the parentheses and multiply by the largest common divisor 2.

= 2(4m + 3)

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 2 and (4m + 3).

Distribute 2 to 4m and 3.

2(4m + 3) = 2(4m) + 2(3)

2(4m + 3) = 8m + 6

When 2 and (4m + 3) are multiplied, we get the given expression (8m + 6).

So, 2 and (4m + 3) are the factors of (8m + 6).

Example 2 :

7y - 14

Solution :

= 7y - 14

The largest common divisor for 7y and 14 is 7.

Divide 7y and 14 by 7.

Write the quotients y and 2 inside the parentheses and multiply by the largest common divisor 7.

= 7(y - 2)

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 7 and (y - 2).

Distribute 7 to y and 2.

7(y - 2) = 7(y) - 7(2)

7(y - 2) = 7y - 14

When 7 and (y - 2) are multiplied, we get the given expression (7y - 14).

So, 7 and (y - 2) are the factors of (7y - 14).

Example 3 :

5a2 - 15a

Solution :

= 5a2 - 15a

The largest common divisor for 5a2 and 15a is 5a.

Divide 5a2 and 15a is 5a.

Write the quotients a and 3 inside the parentheses and multiply by the largest common divisor 5a.

= 5a(a + 3)

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 5a and (a + 3).

Distribute 5a to a and 3.

5a(a + 3) = 5a(a) + 5a(3)

5a(a + 3) = 5a2 + 15a

When 5a and (a + 3) are multiplied, we get the given expression (5a2 + 15a).

So, 5a and (a + 3) are the factors of (5a2 + 15a).

Example 4 :

2y + 6xy

Solution :

= 2y + 6xy

The largest common divisor for 2y and 6xy is 2y.

Divide 2y and 6xy is 2y.

Write the quotients 1 and 3x inside the parentheses and multiply by the largest common divisor 2y.

= 2y(1 + 3x)

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 2y and (1 + 3x).

Distribute 2y to 1 and 3x.

2y(1 + 3x) = 2y(1) + 2y(3x)

2y(1 + 3x) = 2y + 3xy

When 2y and (1 + 3x) are multiplied, we get the given expression (2y + 6xy).

So, 2y and (1 + 3x) are the factors of (2y + 6xy).

Example 5 :

36xy2 - 48x2y

Solution :

= 36xy2 - 48x2y

The largest common divisor for 36 and 48 is 12.

The largest common divisor for x and x2 is x.

The largest common divisor for y2 and y is y.

Therefore, the largest common divisor of 36xy2 and 48x2y is 12xy.

Divide 36xy2 and 48x2y by 12xy.

Write the quotients 3y and 4x inside the parentheses and multiply by the largest common divisor 12xy.

= 12xy(3y - 4x)

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 12xy and (3y - 4x).

Distribute 12xy to 3y and 4x.

12xy(3y - 4x) = 12xy(3y) - 12xy(4x)

12xy(3y - 4x) = 36xy2 - 48x2y

When 12xy and (3y - 4x) are multiplied, we get the given expression 36xy2 - 48x2y.

So, 12xy and (3y - 4x) are the factors of (36xy2 - 48x2y).

Example 6 :

75b2c3 + 60bc6

Solution :

= 75b2c3 + 60bc6

The largest common divisor for 75 and 60 is 15.

The largest common divisor for b2 and b is b.

The largest common divisor for c3 and c6 is c3.

Therefore, the largest common divisor of 75b2c3 and 60bc6 is 15bc3.

Divide 75b2c3 and 60bc6 by 15bc3.

Write the quotients 5b and 4c3 inside the parentheses and multiply by the largest common divisor 15bc3.

= 15bc3(5b + 4c3)

Justify and Evaluate :

To verify our answer, let us use distributive property to multiply 15bc3and (5b + 4c3).

Distribute 12xy to 3y and 4x.

15bc3(5b + 4c3= 15bc3(5b) + 15bc3(4c3)

15bc3(5b + 4c3) = 75b2c3 + 60bc6

When 15bc3 and (5b + 4c3) are multiplied, we get the given expression 75b2c3 + 60bc6.

So, 15bc3and (5b + 4c3) are the factors of (75b2c3 + 60bc6)

## Factoring the Difference of Two Squares

The following algebraic identity can be used to factor the difference of two squares.

a2 - b2 = (a + b)(a - b)

Factor each of the following.

Example 7 :

x2 - 9

Solution :

= x2 - 9

= x2 - 32

= (x + 3)(x - 3)

Example 8 :

4y2 - 25

Solution :

= 4y2 - 25

= 22y2 - 25

= (2y)2 - 52

= (2y + 5)(2y - 5)

Example 9 :

9p2 - 16q2

Solution :

= 9p2 - 16q2

= 32p2 - 42q2

= (3p)2 - (4q)2

= (3p + 4q)(3p - 4q)

Example 10 :

Solution :

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