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 :
5a^{2} - 15a
Solution :
= 5a^{2} - 15a
The largest common divisor for 5a^{2} and 15a is 5a.
Divide 5a^{2} 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) = 5a^{2} + 15a
When 5a and (a + 3) are multiplied, we get the given expression (5a^{2} + 15a).
So, 5a and (a + 3) are the factors of (5a^{2} + 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 :
36xy^{2} - 48x^{2}y
Solution :
= 36xy^{2} - 48x^{2}y
The largest common divisor for 36 and 48 is 12.
The largest common divisor for x and x^{2} is x.
The largest common divisor for y^{2} and y is y.
Therefore, the largest common divisor of 36xy^{2} and 48x^{2}y is 12xy.
Divide 36xy^{2} and 48x^{2}y 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) = 36xy^{2} - 48x^{2}y
When 12xy and (3y - 4x) are multiplied, we get the given expression 36xy^{2} - 48x^{2}y.
So, 12xy and (3y - 4x) are the factors of (36xy^{2} - 48x^{2}y).
Example 6 :
75b^{2}c^{3} + 60bc^{6}
Solution :
= 75b^{2}c^{3} + 60bc^{6}
The largest common divisor for 75 and 60 is 15.
The largest common divisor for b^{2} and b is b.
The largest common divisor for c^{3} and c^{6} is c^{3}.
Therefore, the largest common divisor of 75b^{2}c^{3} and 60bc^{6} is 15bc^{3}.
Divide 75b^{2}c^{3} and 60bc^{6} by 15bc^{3}.
Write the quotients 5b and 4c^{3} inside the parentheses and multiply by the largest common divisor 15bc^{3}.
= 15bc^{3}(5b + 4c^{3})
Justify and Evaluate :
To verify our answer, let us use distributive property to multiply 15bc^{3}and (5b + 4c^{3}).
Distribute 12xy to 3y and 4x.
15bc^{3}(5b + 4c^{3}) = 15bc^{3}(5b) + 15bc^{3}(4c^{3})
15bc^{3}(5b + 4c^{3}) = 75b^{2}c^{3} + 60bc^{6}
When 15bc^{3} and (5b + 4c^{3}) are multiplied, we get the given expression 75b^{2}c^{3} + 60bc^{6}.
So, 15bc^{3}and (5b + 4c^{3}) are the factors of (75b^{2}c^{3} + 60bc^{6})
The following algebraic identity can be used to factor the difference of two squares.
a^{2} - b^{2} = (a + b)(a - b)
Factor each of the following.
Example 7 :
x^{2} - 9
Solution :
= x^{2} - 9
= x^{2} - 3^{2}
= (x + 3)(x - 3)
Example 8 :
4y^{2} - 25
Solution :
= 4y^{2} - 25
= 2^{2}y^{2} - 25
= (2y)^{2} - 5^{2}
= (2y + 5)(2y - 5)
Example 9 :
9p^{2} - 16q^{2}
Solution :
= 9p^{2} - 16q^{2}
= 3^{2}p^{2} - 4^{2}q^{2}
= (3p)^{2} - (4q)^{2}
= (3p + 4q)(3p - 4q)
Example 10 :
Solution :
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