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Division of Polynomials

Using long division to divide polynomials

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Division of Polynomials

Suppose that you know that the area of a rectangular mural in square feet is represented by the polynomial \begin{align*}x^2+2x-24\end{align*}x2+2x24 and that the length of the mural in feet is represented by the binomial \begin{align*}x+6\end{align*}x+6. How would you calculate the width of the mural? Would it also be a binomial? 

Dividing Polynomials

We will begin with a property that is the converse of the Adding Fractions Property presented in previous sections.

The converse of the Adding Fractions Property states that for all real numbers \begin{align*}a, b\end{align*}a,b, and \begin{align*}c\end{align*}c, and \begin{align*}c \neq 0\end{align*}c0, \begin{align*}\frac{a+b}{c}\end{align*}a+bc = \begin{align*}\frac{a}{c}+\frac{b}{c}\end{align*}ac+bc.

This property allows you to separate the numerator into its individual fractions. This property is used when dividing a polynomial by a monomial.

Let's take a look at a couple of problems that use the Adding Fractions Property:

  1. Simplify \begin{align*}\frac{8x^2-4x+16}{2}\end{align*}8x24x+162.

Using the property above, separate the polynomial into its individual fractions.

\begin{align*}&& \frac{8x^2}{2}-\frac{4x}{2}+\frac{16}{2}\\ \text{Reduce.} && 4x^2-2x+8\end{align*}Reduce.8x224x2+1624x22x+8

  1. Simplify \begin{align*}\frac{-3m^2-18m+6}{9m}\end{align*}3m218m+69m.

Separate the trinomial into its individual fractions and reduce.

\begin{align*}& -\frac{3m^2}{9m}-\frac{18m}{9m}+\frac{6}{9m}\\ & -\frac{m}{3}-2+\frac{2}{3m}\end{align*}3m29m18m9m+69mm32+23m

Polynomials can also be divided by binomials. However, instead of separating into its individual fractions, we use a process called long division.

Let's look at the process of polynomial long division by solving the following problem:

Simplify \begin{align*}\frac{x^2+4x+5}{x+3}\end{align*}x2+4x+5x+3.

When we perform division, the expression in the numerator is called the dividend and the expression in the denominator is called the divisor.

To start the division we rewrite the problem in the following form.

Start by dividing the first term in the dividend by the first term in the divisor \begin{align*}\frac{x^2}{x}=x\end{align*}x2x=x. Place the answer on the line above the \begin{align*}x\end{align*}x term.

Next, multiply the \begin{align*}x\end{align*}x term in the answer by each of the \begin{align*}x+3\end{align*}x+3 terms in the divisor and place the result under the divided, matching like terms.

Now subtract \begin{align*}x^2+3x\end{align*}x2+3x from \begin{align*}x^2+4x+5\end{align*}x2+4x+5. It is useful to change the signs of the terms of \begin{align*}x^2+3x\end{align*}x2+3x to \begin{align*}-x^2-3x\end{align*}x23x and add like terms vertically.

Now, bring down 5, the next term in the dividend.

Repeat the process. First divide the first term of \begin{align*}x+5\end{align*}x+5 by the first term of the divisor \begin{align*}\left(\frac{x}{x}\right)=1\end{align*}. Place this answer on the line above the constant term of the dividend.

Multiply 1 by the divisor \begin{align*}x+3\end{align*} and write the answer below \begin{align*}x+5\end{align*}, matching like terms.


Subtract \begin{align*}x+3\end{align*} from \begin{align*}x+5\end{align*} by changing the signs of \begin{align*}x+3\end{align*} to \begin{align*}-x-3\end{align*} and adding like terms.

Since there are no more terms from the dividend to bring down, we are done.

The answer is \begin{align*}x+1\end{align*} with a remainder of 2.


Example 1

Earlier, you were asked to calculate the width of a rectangular mural. You know that the area of the mural in square feet is represented by \begin{align*}x^2+2x-24\end{align*}, and that the length in square feet is represented by \begin{align*}x+6\end{align*}.

We know that the area of a rectangle is equal to it's width times the length. Let \begin{align*}w=\end{align*} the width and \begin{align*}l=\end{align*} the length. To find the width, we can plug what we know into this equation and solve for w. This will involve dividing. We could use long division to solve this problem. However, we can also use factoring to find a common factor to divide out. This is the process that we will follow. 

\begin{align*}& Area=wl\\ & Plug\ and\ chug \rightarrow x^2-2x-24=(w)(x+6)\\ & Solve\ for\ w\ \rightarrow w=\frac{x^2-2x-24}{x+6}\\ & Factor\ the \ numerator\rightarrow w= \frac{(x-8)(x+6)}{x+6}\\ & Take\ out\ common\ factor \rightarrow w=x-8 \end{align*}

Example 2

Divide \begin{align*}9x^2-16\end{align*} by \begin{align*}3x+4\end{align*}.

You are being asked to simplify:

\begin{align*} \frac{9x^2-16}{3x+4}.\end{align*}

You could use long division to find the answer. You can also use patterns of polynomials to simplify and cancel.

Recall that \begin{align*}a^2-b^2=(a+b)(a-b).\end{align*} Use this pattern to solve this problem since \begin{align*} 9x^2-16=(3x)^2-4^2\end{align*}:

\begin{align*} \frac{9x^2-16}{3x+4}&=\frac{(3x)^2-4^2}{3x+4}\\ &=\frac{(3x-4)(3x+4)}{3x+4}\\ &=3x-4 \end{align*}


Divide the following polynomials.

  1. \begin{align*}\frac{2x+4}{2}\end{align*}
  2. \begin{align*}\frac{x-4}{x}\end{align*}
  3. \begin{align*}\frac{5x-35}{5x}\end{align*}
  4. \begin{align*}\frac{x^2+2x-5}{x}\end{align*}
  5. \begin{align*}\frac{4x^2+12x-36}{-4x}\end{align*}
  6. \begin{align*}\frac{2x^2+10x+7}{2x^2}\end{align*}
  7. \begin{align*}\frac{x^3-x}{-2x^2}\end{align*}
  8. \begin{align*}\frac{5x^4-9}{3x}\end{align*}
  9. \begin{align*}\frac{x^3-12x^2+3x-4}{12x^2}\end{align*}
  10. \begin{align*}\frac{3-6x+x^3}{-9x^3}\end{align*}
  11. \begin{align*}\frac{x^2+3x+6}{x+1}\end{align*}
  12. \begin{align*}\frac{x^2-9x+6}{x-1}\end{align*}
  13. \begin{align*}\frac{x^2+5x+4}{x+4}\end{align*}
  14. \begin{align*}\frac{x^2-10x+25}{x-5}\end{align*}
  15. \begin{align*}\frac{x^2-20x+12}{x-3}\end{align*}
  16. \begin{align*}\frac{3x^2-x+5}{x-2}\end{align*}
  17. \begin{align*}\frac{9x^2+2x-8}{x+4}\end{align*}
  18. \begin{align*}\frac{3x^2-4}{3x+1}\end{align*}
  19. \begin{align*}\frac{5x^2+2x-9}{2x-1}\end{align*}
  20. \begin{align*}\frac{x^2-6x-12}{5x+4}\end{align*}
  21. \begin{align*}\frac{x^4-2x}{8x+24}\end{align*}
  22. \begin{align*}\frac{x^3+1}{4x-1}\end{align*}

Mixed Review

  1. Boyle’s Law states that the pressure of a compressed gas varies inversely as its pressure. If the pressure of a 200-pound gas is 16.75 psi, find the pressure if the amount of gas is 60 pounds.
  2. Is \begin{align*}5x^3+x^2-x^{-1}+8\end{align*} an example of a polynomial? Explain your answer.
  3. Find the slope of the line perpendicular to \begin{align*}y=-\frac{3}{4} x+5\end{align*}.
  4. How many two-person teams can be made from a group of nine individuals?
  5. Solve for \begin{align*}m: -4= \frac{\sqrt{m-3}}{-2}\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3. 

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Adding Fraction Property

For all real numbers a, b, and c, and c \neq 0, \frac{a+b}{c} = \frac{a}{c}+\frac{b}{c}.


The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. \frac{5}{8} has denominator 8.


In a division problem, the dividend is the number or expression that is being divided.


In a division problem, the divisor is the number or expression that is being divided into the dividend. For example: In the expression 152 \div 6, 6 is the divisor and 152 is the dividend.

Polynomial long division

Polynomial long division is the standard method of long division, applied to the division of polynomials.

Rational Expression

A rational expression is a fraction with polynomials in the numerator and the denominator.

Rational Root Theorem

The rational root theorem states that for a polynomial, f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, where a_n, a_{n-1}, \cdots a_0 are integers, the rational roots can be determined from the factors of a_n and a_0. More specifically, if p is a factor of a_0 and q is a factor of a_n, then all the rational factors will have the form \pm \frac{p}{q}.

Remainder Theorem

The remainder theorem states that if f(k) = r, then r is the remainder when dividing f(x) by (x - k).

Synthetic Division

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.

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