Recall that the Distributive Property states that
ab + ac = a(b + c)
The Distributive Property allows you to factor out the GCF of the terms in a polynomial to write a factored form of the polynomial.
A polynomial is in its factored form when it is written as a product of monomials and polynomials that cannot be factored further.
The expression 2(3x - 4x) is not fully factored because the terms in the parentheses have a common factor of x.
Factor each polynomial. Check your answer.
Example 1 :
4x2 - 3x
Solution :
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)
Check :
Multiply to check your answer.
x(4x - 3) = x(4x) - x(3)
= 4x2 - 3x
The product is the original polynomial.
Example 2 :
10y3 + 20y2 - 5y
Solution :
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)
Check :
Multiply to check your answer.
5y(2y2 + 4y - 1) = 5y(2y2) + 5y(4y) + 5y(-1)
= 10y3 + 20y2 - 5y
The product is the original polynomial.
Example 3 :
-12a - 8a2
Solution :
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)
Check :
Multiply to check your answer.
-4a(3 + 2a) = -4a(3) - 4a(2a)
= -12a - 8a2
The product is the original polynomial.
Example 4 :
5x2 + 7
Solution :
Find the GCF :
5x2 = 5 ⋅ x ⋅ x
7 = 7
There are no common factors other than 1.
The polynomial cannot be factored.
Sometimes the GCF of terms is a binomial. This GCF is called a common binomial factor. You factor out a common binomial factor the same way you factor out a monomial factor.
Factor each expression.
Example 5 :
7(x - 3) - 2x(x - 3)
Solution :
(x - 3) is a common binomial factor.
= 7(x - 3) - 2x(x - 3)
Factor out (x - 3).
= (x - 3)(7 - 2x)
Example 6 :
-y(y2 + 5) + (y2 + 5)
Solution :
(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)
Example 7 :
9n(n + 4) - 5(4 + n)
Solution :
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)
Example 8 :
-3y2(y + 2) + 4(y - 7)
Solution :
There are no common factors.
The expression cannot be factored.
You may be able to factor a polynomial by grouping. When a polynomial has four terms, you can make two groups and factor out the GCF from each group.
Factor each polynomial by grouping. Check your answer.
Example 9 :
12x3 - 9x2 + 20x - 15
Solution :
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)
Check :
Multiply to check your answer.
(4x - 3)(3x2 + 5) = 4x(3x2) + 4x(5) - 3(3x2) - 3(5)
= 12x3 + 20x - 9x2 - 15
= 12x3 - 9x2 + 20x - 15
The product is the original polynomial.
Example 10 :
9a3 + 18a2 + a + 2
Solution :
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)
Check :
Multiply to check your answer.
(a + 2)(9a2 + 1) = a(9a2) + a(1) + 2(9a2) + 2(1)
= 9a3 + a + 18a2 + 2
= 9a3 + 18a2 + a + 2
The product is the original polynomial.
Recognizing opposite binomials can help you factor polynomials. The binomials (5 - x) and (x - 5) are opposites. Notice (5 - x) can be written as -1(x - 5).
= -1(x - 5)
Distributive Property.
= (-1)(x) + (-1)(-5)
Simplify.
= -x + 5
Commutative Property of Addition.
= 5 - x
So,
5 - x = -1(x - 5)
Example 11 :
Factor 3x3 - 15x2 + 10 - 2x by grouping.
Solution :
= 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)
Example 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.
Solution :
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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