Multiplying Monomials by Polynomials

Multiplication of Monomials by Polynomials

Read the information. It may be helpful to review the section Multiplying Monomials at the beginning of the page. Then, focus on the section Using the Distributive Property and the examples that follow it. After you have reviewed the material, complete review problems 6 to 11 and check your answers.

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/
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