Multiply Polynomials
The key to multiplying polynomials is the Distributive Property.
a(b + c) = a·b + a·c
It doesn’t matter how small or large the polynomial is, we will always
distribute a monomial through the parentheses, multiplying a monomial
by a monomial. We’ve already multiplied monomials when we
discussed exponent rules.
Multiplying monomials is done by
multiplying the coefficients and then adding the exponents (product rule)
on like variable factors.
Example 1: Multiply: (4x3y4z)(2x2y6z3)
4·2·x3+2·y4+6·z1+3
8x5y10z4
Multiply coefficients and add exponents
Example 2: Multiply: 4x3(5x2 – 2x + 5)
4x3(5x2) + 4x3(-2x) + 4x3(5) Distribute 4x3
4·5x3+2 + 4(-2)x3+1 + 4·5x3 Multiply coefficients and add exponents
20x5 – 8x4 – 20x3
Example 3: Multiply: 2a3b(3ab2 – 4a)
2a3b(3ab2) + 2a3b(4a)
2·3a3+1b1+2 + 2(-4)a3+1b
6a4b3 – 8a4b
Distribute 2a3b
Multiply coefficients and add exponents
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
To multiply by a binomial, distribute the first term (monomial), then
distribute the second term (monomial), and finally combine like terms.
Example 4: Multiply: (4x + 7y)(3x – 2y)
4x(3x – 2y) + 7y(3x – 2y)
Distribute 4x first and then distribute 7y
4x(3x) + 4x(-2y) + 7y(3x) + 7y(-2y)
12x2 – 8xy + 21xy – 14y2
Multiply coefficients and add exponents
2
2
12x + 13xy – 14y
Combine like terms
Example 5: Multiply: (5x – 6)(4x – 1)
5x(4x – 1) – 6(4x – 1)
Distribute 5x first and then distribute -6
5x(4x) + 5x(-1) – 6(4x) – 6(-1)
20x2 – 5x – 24x + 6
Multiply coefficients and add exponents
2
20x – 29x + 6
Combine like terms
Example 6: Multiply: (x – 5)(x + 5)
x(x + 5) – 5(x + 5)
x(x) + x(5) – 5(x) – 5(5)
x2 + 5x – 5x + 25
x2 – 25
Distribute x first and then distribute -5
Multiply coefficients and add exponents
Combine like terms
Notice in the above example 6 that when we combined like terms, the
two middle terms added to zero. This will always happen when
multiplying two binomials that are the same terms with one binomial
being a sum and the other a difference. A shortcut is to merely multiply
the first terms of each binomial and multiply the last terms of each,
resulting in the difference of two squares.
(a + b)(a – b) = a·a – b·b = a2 – b2
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 7: Multiply: (3x + 7)(3x – 7)
Using the above formula for the difference of two squares:
3x·3x – 7·7
9x2 – 49
Example 8: Multiply: (2x + 5)2
The outside exponent indicates two factors of (2x + 5)
(2x + 5)(2x + 5)
2x(2x + 5) + 5(2x + 5)
Distribute 2x first and then distribute 5
2x(2x) + 2x(5) + 5(2x) + 5(5)
4x2 + 10x + 10x + 25
Multiply coefficients and add exponents
2
4x + 20x + 25
Combine like terms
When multiplying a binomial squared, as in the above example 8, we
can either write two identical binomials and use the Distributive
Property twice or follow the shortcut given below. Notice in the above
example 8 that to achieve the final product, the first term is squared; the
middle term is the first term times the last time twice or 2 times the first
and last; and the last term is squared.
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Example 9: Multiply: (3x – 7y)2
Using the above formula for squaring a binomial:
(3x)2 – 2(3x)(7y) + (7y)2
9x2 – 42xy + 49y2
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
A common error is to distribute the square through the parentheses:
(x+ 5)2 = x2 + 25. This doesn’t work, because it is missing the middle
term 10x. This is why it is important to either use the above shortcut or
write two identical binomials and use the Distributive Property twice,
whichever you are more comfortable with.
When one of the factors is larger than a binomial, we still use the
Distributive Property – just more often. Adding vertically by putting
like terms in columns works well with larger polynomials.
Example 10: Multiply: (2x – 5)(4x2 – 7x + 3)
2x(4x2 – 7x + 3) - 5(4x2 – 7x + 3) Distribute 2x first and then distribute -5
2x(4x2) + 2x(-7x) + 2x(3)
– 5(4x2) – 5(-7x) – 5(3)
Multiply coefficients and add exponents
8x3 – 14x2 + 6x
– 20x2 + 35x – 15
8x3 – 34x2 + 41x – 15
Put like terms in columns and combine
Example 11: Multiply: (2a2 + 6a + 3)(7a2 – 6a + 1)
2a2(7a2 – 6a + 1) + 6a(7a2 – 6a + 1) + 3(7a2 – 6a + 1) Distribute 2a2, 6a, 3
2a2(7a2) + 2a2(-6a) + 2a2(1)
+ 6a(7a2) + 6a(-6a) + 6a(1)
+ 3(7a2) + 3(-6a) + 3(1)
14a4 – 12a3 + 2a2
+ 42a3 – 36a2 + 6a
+ 21a2 – 18a + 3
14a4 + 30a3 – 13a2 – 12a + 3
Multiply coefficients and add exponents
Put like terms in columns and combine
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)