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Wize Grade 11 Mathematics Textbook > Polynomials and Operations

Dividing Polynomials

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Dividing Polynomials

To divide polynomials we can use a process similar to dividing numbers.

  1. The polynomial being divide goes into the division bar
  2. The polynomial doing the dividing goes outside
  3. We divide one term at a time
  4. The process stops when the power of the remaining polynomial is smaller than dividing polynomial

Example 1

Divide the following polynomials

6x3+5x2−8x−33x+1\displaystyle \frac{6x^3+5x^2-8x-3}{3x+1}



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Missing terms

If one of the polynomials seems to be missing a power of x, you can put in a place holder.
This is done by putting a zero with the missing power of x.



Example 2

Divide the following polynomials

27x3−13x−1\displaystyle \frac{27x^3 - 1}{3x - 1}

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Synthetic Division

A faster way to divide polynomials is the process of synthetic division. In this process we only need to keep track of the coefficients.
The simplest version of synthetic division only allows for diving by linear polynomials.


  1. The coefficients of the polynomial being divided go into division bar
  2. The value of x that makes the dividing polynomial zero goes to the outside
  3. The first number is copied down
  4. We multiply by the dividing number and copy down the result
  5. Columns are added and we repeat step 4 for the next set of numbers
  6. The numbers along the bottom form the coefficients of the result


Example 1

Divide the following polynomials.

3x4+13x3−7x2+19x+20x+5\displaystyle \frac{3x^4+13x^3-7x^2+19x+20}{x+5}


=3x3−2x2+3x+4=3x^3 - 2x^2 + 3x + 4
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Example: Dividing Polynomials

The volume of a cylinder can can be expressed by the formula
V=Ah\boxed{V = A h}
Where AA is the area of the circle on the end, and hh is the height.


Suppose the volume of a given can of beans is given by

7πx3+32πx2−15πx+π 7\pi x^3 + 32\pi x^2 - 15\pi x + \pi

If the can is (x+5)(x + 5) cm tall, what is the expression that represents the area of the lid?

Using the formula for a cylinder we can write this as

7πx3+32πx2−15πx+π=A(x+5)7\pi x^3 + 32 \pi x^2 - 15 \pi x + \pi = A (x + 5)

Factoring out the π\pi and dividing by (x+5)(x + 5) this becomes

(7x3+32x2−15x+1)πx+5=A\displaystyle \frac{(7x^3 + 32x^2 - 15x + 1)\pi}{x+5} = A

Dividing the two polynomials on the left this simplifies to

(7x2−3x+1)π=A7πx2−3πx+π=A\begin{aligned} (7x^2 - 3x + 1)\pi &= A \\ 7 \pi x^2 - 3\pi x + \pi &= A \end{aligned}

This represents the area of the lid.

Practice: Dividing Polynomials

Felix was working on dividing the following polynomials using synthetic division, but he is not quite sure how to interpret the result.

3x2−5x+7x−2\displaystyle\frac{3x^2 - 5x + 7}{x - 2}



Select the polynomial below that represents the correct result.

Practice: Dividing Polynomials

Divide the following polynomials using any method

1. 8x4+2x3−18x2+21x−94x−3\displaystyle \frac{8x^4 + 2x^3 - 18x^2 + 21x -9}{4x - 3}


2. x3−16x+21x−3\displaystyle \frac{x^3 - 16x + 21}{x - 3}


Practice: Dividing Polynomials

The following polynomial x+5x + 5 was divided into another polynomial to produce the following result

?x+5=7x2−x+3+1x+5\displaystyle \frac{?}{x+5} = 7x^2 - x + 3 + \frac{1}{x+5}
Find the polynomial that was originally divided.
Extra Practice
Polynomial Functions
Divide 2x4−11x3+12x2+9x−22x^4-11x^3+12x^2+9x-2 by 2x+12x+1.
Dividing Polynomials
Practice: Dividing Polynomials
Divide 5x4+2x3−9x2+x−1x2+x−1\displaystyle\frac{5x^4+2x^3-9x^2+x-1}{x^2+x-1} and choose the correct quotient.
Dividing Polynomials
Practice: Dividing Polynomials
Divide 2x3+17x+3x2−5x\frac{2x^3+17x+3}{x^2-5x} and choose the correct quotient.
Dividing Polynomials
Practice: Dividing Polynomials
Determine the division statement:
4x3−5x2+8x−3\dfrac{4x^3-5x^2+8}{x-3}
Dividing Polynomials
Practice: Dividing Polynomials
Determine the division statement:
−x5−4x4+2x2+xx−1\dfrac{-x^5-4x^4+2x^2+x}{x-1}
Dividing Polynomials
Practice: Dividing Polynomials
Find the remainder using synthetic division when (−5x4+3x3−2x2+6) ÷ (x−3)(-5x^4+3x^3-2x^2+6)~\div~(x-3)