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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

Multiplication of 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.

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.


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*}


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, we have to 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*}


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



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 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*}

Review (Answers)

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

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