HOW TO FACTOR BINOMIALS

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)

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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