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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, 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 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*}

\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*}

### Example

#### 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*}

### Review

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*}

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