Multiplying Monomials by Polynomials

Just as we can add and subtract polynomials, we can also multiply them. The Distributive Property and the techniques you’ve learned for dealing with exponents will be useful here.

When multiplying polynomials, remember the exponent rules, particularly the product rule: \(\begin{align*}x^n \cdot x^m=x^{n+m}\end{align*}\).

If the expressions have coefficients and more than one variable, multiply the coefficients just as you would any number and apply the product rule to each variable separately.

Multiplying Monomials 

Multiply the following monomials.

a) \(\begin{align*}(2x^2)(5x^3)\end{align*}\)

\(\begin{align*}(2x^2)(5x^3)=(2 \cdot 5) \cdot (x^2 \cdot x^3)=10x^{2+3} = 10x^5\end{align*}\)

b) \(\begin{align*}(-3y^4)(2y^2)\end{align*}\)

\(\begin{align*}(-3y^4)(2y^2)=(-3 \cdot 2) \cdot (y^4 \cdot y^2)=-6y^{4+2}=-6y^6\end{align*}\)

c) \(\begin{align*}(3xy^5)(-6x^4y^2)\end{align*}\)

\(\begin{align*}(3xy^5)(-6x^4y^2)=-18x^{1+4}y^{5+2}=-18x^5y^7\end{align*}\)

d) \(\begin{align*}(-12a^2b^3c^4)(-3a^2b^2)\end{align*}\)

\(\begin{align*}(-12a^2b^3c^4)(-3a^2b^2)=36a^{2+2}b^{3+2}c^4 = 36a^4b^5c^4\end{align*}\)

To multiply a polynomial by a monomial, use the Distributive Property. Remember, that property says that \(\begin{align*}a(b + c) = ab + ac\end{align*}\).

Using the Distributive Property 

1. Multiply:

a) \(\begin{align*}3(x^2+3x-5)\end{align*}\)

\(\begin{align*}3(x^2+3x-5)=3(x^2)+3(3x)-3(5)=3x^2+9x-15\end{align*}\)

b) \(\begin{align*}4x(3x^2-7)\end{align*}\)

\(\begin{align*}4x(3x^2-7)=(4x)(3x^2)+(4x)(-7)=12x^3-28x\end{align*}\)

c) \(\begin{align*}-7y(4y^2-2y+1)\end{align*}\)

\(\begin{align*}-7y(4y^2-2y+1)&=(-7y)(4y^2)+(-7y)(-2y)+(-7y)(1)\\ &=-28y^3+14y^2-7y\end{align*}\)

Notice that when you use the Distributive Property, the problem becomes a matter of just multiplying monomials by monomials and adding all the separate parts together.

2. Multiply:

a) \(\begin{align*}2x^3(-3x^4+2x^3-10x^2+7x+9)\end{align*}\)

\(\begin{align*}2x^3(-3x^4+2x^3-10x^2+7x+9)&=(2x^3)(-3x^4)+(2x^3)(2x^3)+(2x^3)(-10x^2)+(2x^3)(7x)+(2x^3)(9)\\ & = -6x^7+4x^6-20x^5+14x^4+18x^3\end{align*}\)

b) \(\begin{align*}-7a^2bc^3(5a^2-3b^2-9c^2)\end{align*}\)

\(\begin{align*}-7a^2bc^3(5a^2-3b^2-9c^2) & = (-7a^2bc^3)(5a^2)+(-7a^2bc^3)(-3b^2)+(-7a^2bc^3)(-9c^2)\\ & = -35a^4bc^3 + 21a^2b^3c^3 + 63a^2bc^5\end{align*}\)

Source: cK-12, https://www.ck12.org/algebra/multiply-polynomials-by-monomials/lesson/Multiplication-of-Monomials-by-Polynomials-ALG-I/
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 License.

Example 1

Multiply \(\begin{align*}-2a^2b^4(3ab^2+7a^3b-9a+3)\end{align*}\).

Multiply the monomial by each term inside the parenthesis:

\(\begin{align*}& -2a^2b^4(3ab^2+7a^3b-9a+3)\\ & = (-2a^2b^4)(3ab^2)+(-2a^2b^4)(7a^3b)+(-2a^2b^4)(-9a)+(-2a^2b^4)(3)\\ & = -6a^3b^6-14a^5b^5+18a^5b^4-6a^2b^4\end{align*}\)

Multiply and simplify.

6. \(\begin{align*}17(8x-10)\end{align*}\)

7. \(\begin{align*}2x(4x-5)\end{align*}\)

8. \(\begin{align*}9x^3(3x^2-2x+7)\end{align*}\)

9. \(\begin{align*}3x(2y^2+y-5)\end{align*}\)

10. \(\begin{align*}10q(3q^2r+5r)\end{align*}\)

11. \(\begin{align*}-3a^2b(9a^2-4b^2)\end{align*}\)

6. \(17(8 x-10)=134 x-170\)

7. \(2 x(4 x-5)=8 x^{2}-10 x\)

8. \(9 x^{3}\left(3 x^{2}-2 x+7\right)=27 x^{5}-18 x^{4}+63 x^{3}\)

9. \(3 x\left(2 y^{2}+y-5\right)=6 x y^{2}+3 x y-15 x\)

10. \(10 q\left(3 q^{2} r+5 r\right)=30 q^{3} r+50 q r\)

11. \(-3 a^{2} b\left(9 a^{2}-4 b^{2}\right)=-27 a^{4} b+12 a^{2} b^{3}\)