8.5: Dividing Polynomials by a Monomial

Learning Outcomes

Divide monomials
Divide polynomials by monomials

Key words

Quotient: the result of dividing
Quotient Property of Exponents: to divide two terms with the same base, subtract the exponents and keep the common base
Dividend: the expression being divided
Divisor: the expression dividing into the dividend

Dividing by a Monomial

In a previous chapter we learned about the properties of exponents. In particular, we learned that to divide two terms with the same base, we subtract the exponents and keep the common base:

$\frac{x^m}{x^n}=x^{m-n}$

We will now use this quotient property of exponents to divide two monomials.

Example

Find the quotient:

$x^7\div x^4$
$\frac{y^5}{y^4}$
$\frac{n^8}{n^8}$
$\frac{z^2}{z^5}$

Solution

$x^7\div x^4=x^{7-4}=x^3$ Keep the common base, $x$ , and subtract the exponents.
$\frac{y^5}{y^4}=y^{5-4}=y^1=y$ Keep the common base, $y$ , and subtract the exponents.
$\frac{n^8}{n^8}=n^{8-8}=n^0=1$ Remember that $x^0=1$ for all $x\ne0$
$\frac{z^2}{z^5}=z^{2-5}=z^{-3}=\frac{1}{z^3}$ Remember that a negative exponent on the numerator becomes a positive exponent on the denominator: $x^{-n}=\frac{1}{n}$

Try It

Find the quotient:

$x^6\div x^2$
$\frac{y^8}{y^5}$
$\frac{n^3}{n^3}$
$\frac{z^4}{z^7}$

Show Answer

$x^6\div x^2=x^4$
$\frac{y^8}{y^5}=y^3$
$\frac{n^3}{n^3}=1$
$\frac{z^4}{z^7}=\frac{1}{z^3}$

Technically, any divisor cannot equal zero, as division by zero is undefined. Also, if we get an answer of $x^0$ , the answer will always be $1$ , provided that $x\ne0$ . $0^0$ is undefined. Notice that dividing two monomials does not always result in a monomial. For example, $\frac{z^4}{z^7}=\frac{1}{z^3}$ does not result in a monomial. Remember that monomials cannot have negative exponents; the exponent must be a whole number.

When there are coefficients attached to the variables, we divide the coefficients and divide the variables.

EXAMPLE

Find the quotient: $56{x}^{5}\div 7{x}^{2}$

Solution

	$56{x}^{5}\div 7{x}^{2}$
Rewrite as a fraction.	${\dfrac{56{x}^{5}}{7{x}^{2}}}$
Use fraction multiplication to separate the number part from the variable part.	${\dfrac{56}{7}}\cdot {\dfrac{{x}^{5}}{{x}^{2}}}$
Use the Quotient Property: keep the base, subtract the exponents	$8{x}^{3}$

Answer

$56{x}^{5}\div 7{x}^{2}=8{x}^{3}$

TRY IT

1. Find the quotient: $63{x}^{8}\div 9{x}^{4}$

Show Solution

$7{x}^{4}$

2. Find the quotient: $96{y}^{11}\div 6{y}^{8}$

Show Solution

$16{y}^{3}$

Try It

Dividing a Polynomial by a Monomial

The distributive property states that we can distribute a factor that is being multiplied by a sum or difference: $a(b+c)=ab+ac$ . If the term being multiplied is a fraction, $\frac{1}{a}$ , the distributive property tells us:

$\begin{equation}\begin{aligned}&\;\;\;\; \\ \frac{1}{a}(b+c)&=\frac{1}{a}\cdot b+\frac{1}{a}\cdot c\end{aligned}\end{equation}$

But, $\frac{1}{a}(b+c)=\frac{b+c}{a}$ and $\frac{1}{a}\cdot b+\frac{1}{a}\cdot c=\frac{b}{a}+\frac{c}{a}$ by multiplication of fractions.

So, $\frac{b+c}{a}=\frac{b}{a}+\frac{c}{a}$

In other words, we can distribute a divisor that is being divided into a sum or difference.

In this arithmetic example, we can add all the terms in the numerator, then divide by $2$ .

$\frac{\text{dividend}\rightarrow}{\text{divisor}\rightarrow}\,\,\,\,\,\, \frac{8+4+10}{2}=\frac{22}{2}=11$

Or we can first divide each term by $2$ , then simplify the result.

$\frac{8}{2}+\frac{4}{2}+\frac{10}{2}=4+2+5=11$

Either way gives the same result. The second way is helpful when we can’t combine like terms in the numerator, as in a polynomial divided by a monomial.

Example

Divide. $\frac{9a^3+6a}{3a^2}$

Solution

Distribute $3a^2$ over the polynomial by dividing each term by $3a^2$ :

$\frac{9a^3}{3a^2}+\frac{6a}{3a^2}$

Divide each term, a monomial divided by another monomial:

$\begin{array}{c}3a^{3-2}+2a^{1-2}\\\text{ }\\=3a^{1}+2a^{-1}\\\text{ }\\=3a+2a^{-1}\end{array}$

Rewrite $a^{-1}$ with positive exponents, as a matter of convention:

$3a+2a^{-1}=3a+\frac{2}{a}$

Answer

$\frac{9a^3+6a}{3a^2}=3a+\frac{2}{a}$

The distributive property can be extended to any number of terms, so the next example applies the same ideas to divide a trinomial by a monomial. We can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. Remember that simplifying with exponents includes rewriting negative exponents in the numerator as positive exponents in the denominator. Also remember to pay attention to the signs of the terms.

Example

Divide. $\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}$

Solution

Divide each term in the polynomial by the monomial:

$\frac{27{{y}^{4}}}{-6y}+\frac{6{{y}^{2}}}{-6y}-\frac{18}{-6y}$

Note how the term $-\frac{18}{-6y}$ does not have a $y$ in the numerator, so division is only applied to the numbers $18$ and $-6$ . Also, 27 doesn’t divide exactly by $-6$ , so we are left with a fraction as the coefficient on the $y^3$ term.

Simplify:

$-\frac{9}{2}{{y}^{3}}-y+\frac{3}{y}$

Answer

$\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\frac{9}{2}{{y}^{3}}-y+\frac{3}{y}$

Try It

No matter the number of terms in the polynomial, we can use the distributive property to divide by a monomial.

Example

Divide $6x^6-3x^4+9x^2-7$ by $-3x^3$

Solution

Write as division:

$\frac{6x^6-3x^4+9x^2-7}{-3x^3}$

Distribute the monomial to each term in the polynomial:

$\frac{6x^6}{-3x^3}-\frac{3x^4}{-3x^3}+\frac{9x^2}{-3x^3}-\frac{7}{-3x^3}$

Simplify:

$-2x^3+x-3x^{-1}+\frac{7}{3x^3}$

Write the negative exponent as a positive exponent on the denominator:

$-2x^3+x-\frac{3}{x}+\frac{7}{3x^3}$

Answer

$\frac{6x^6-3x^4+9x^2-7}{-3x^3}=-2x^3+x-\frac{3}{x}+\frac{7}{3x^3}$

Example

Divide: $\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}$

Solution

$\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}$

Distribute $-12x^2$ to each term of the polynomial:

$\frac{24x^8}{-12x^2}+\frac{36x^7}{-12x^2}-\frac{-12x^4}{-12x^2}-\frac{-60x^3}{-12x^2}+\frac{6x}{-12x^2}$

Simplify:

$-2x^6-3x^5+x^2+5x-\frac{1}{2}x^{-1}$

Write the negative exponent as a positive exponent on the denominator:

$-2x^6-3x^5+x^2+5x-\frac{1}{2x}$

Answer

$\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}=-2x^6-3x^5+x^2+5x-\frac{1}{2x}$

Try It

1. Divide: $\frac{36x^6-16x^4-24x^3+6x}{-4x^2}$

Show Answer

- 9 x^{4} + 4 x^{2} + 6 x - \frac{3}{2 x}

$-9x^4+4x^2+6x-\frac{3}{2x}$

2. Divide: $\frac{42x^7-14x^5+21x^4-35x^3+6x}{7x^3}$

Show Answer

6 x^{4} - 2 x^{2} + 3 x - 5 + \frac{6}{7 x^{2}}

$6x^4-2x^2+3x-5+\frac{6}{7x^2}$

3. Divide: $\frac{-42x^9-24x^7+9x^4-36x^3-x}{-6x^4}$

Show Answer

7 x^{5} + 4 x^{3} - \frac{3}{2} + \frac{6}{x} + \frac{1}{6 x^{3}}

$7x^5+4x^3-\frac{3}{2}+\frac{6}{x}+\frac{1}{6x^3}$

Chapter 8: Polynomials