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Multiplication of Monomials by Polynomials

Distribute single terms by multiplying them by other terms

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Multiplication of Monomials by Polynomials

What if you had a monomial and polynomial like \begin{align*}3x^3\end{align*} and \begin{align*}x^2 + 4\end{align*}? How could you multiply them? After completing this Concept, you'll be able multiply a polynomial by a monomial.

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CK-12 Foundation: Multiplying a Polynomial by a Monomial


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.

Multiplying a Polynomial by a Monomial

When multiplying polynomials, we must remember the exponent rules that we learned in the last chapter. Especially important is the product rule: \begin{align*}x^n \cdot x^m=x^{n+m}\end{align*}.

If the expressions we are multiplying have coefficients and more than one variable, we multiply the coefficients just as we would any number and we apply the product rule on each variable separately.

Example A

Multiply the following monomials.

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

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

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

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


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

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

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

Example B


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

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

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


a) \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)=(4x)(3x^2)+(4x)(-7)=12x^3-28x\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 we use the Distributive Property, the problem becomes a matter of just multiplying monomials by monomials and adding all the separate parts together.

Example C


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

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


a) \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) & = (-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*}

Watch this video for help with the Examples above.

CK-12 Foundation: Multiplying a Polynomial by a Monomial

Guided Practice

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*}

Explore More

Multiply the following monomials.

  1. \begin{align*}(2x)(-7x)\end{align*}
  2. \begin{align*}(10x)(3xy)\end{align*}
  3. \begin{align*}(4mn)(0.5nm^2)\end{align*}
  4. \begin{align*}(-5a^2b)(-12a^3b^3)\end{align*}
  5. \begin{align*}(3xy^2z^2)(15x^2yz^3)\end{align*}

Multiply and simplify.

  1. \begin{align*}17(8x-10)\end{align*}
  2. \begin{align*}2x(4x-5)\end{align*}
  3. \begin{align*}9x^3(3x^2-2x+7)\end{align*}
  4. \begin{align*}3x(2y^2+y-5)\end{align*}
  5. \begin{align*}10q(3q^2r+5r)\end{align*}
  6. \begin{align*}-3a^2b(9a^2-4b^2)\end{align*}

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.3. 

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