Dividing with Polynomials

(long division with polynomials & how to find the quotient and remainder functions)


We now learn how to divide with polynomials, in particular we learn about long division with polynomials. Just as with long division with whole numbers, when we divide one polynomial by another we obtain a quotient and a remainder; we'll speak of quotient and remainder polynomial functions.


For instance, in the same way that we can write: \[227 \div 7 = \underbrace{32}_{\text{quotient}} + \frac{\overbrace{3}^{\text{remainder}}}{7}\] by the end of this section, we'll be comfortable in showing that we can write: \[\begin{pmatrix}x^5 - 3x^4+5x^2+8 \end{pmatrix} \div \begin{pmatrix} x^2 - x +1 \end{pmatrix} = \underbrace{x^3 - 2x^2 - 3x + 4}_{\text{quotient } Q(x)} + \frac{\overbrace{7x+4}^{\text{remainder } R(x)}}{x^2 - x +1}\] We start by learning about Euclidean Division of Polynomials and then learn the method for long division with polynomials.

Euclidean Division of Polynomials


Given a polynomial function \(f(x)\) of degree \(n\) and a second polynomial \(g(x)\) of degree \(m\), such that \(m\leq n\), meaning: \[deg \begin{bmatrix}g(x) \end{bmatrix} \leq deg \begin{bmatrix} f(x) \end{bmatrix} \] there exist two unique polynomial functions \(Q(x)\) and \(R(x)\) such that: \[f(x) = g(x).Q(x)+R(x)\] Where \(deg \begin{bmatrix}R(x)\end{bmatrix}< deg \begin{bmatrix}g(x)\end{bmatrix}\) and :

  • \(Q(x)\) is known as the quotient polynomial
  • \(R(x)\) is known as the remainder polynomial.
We can rearrange \(f(x) = g(x).Q(x)+R(x)\), dividing both sides by \(g(x)\), to write it as: \[\frac{f(x)}{g(x)} = Q(x)+\frac{R(x)}{g(x)}\] Where the terms quotient and remainder appear to have more meaning, since they're obtained by dividing \(f(x)\) by \(g(x)\).


Method for Dividing Polynomials

When we write \(\frac{f(x)}{g(x)} = Q(x)+\frac{R(x)}{g(x)}\) it highlights the fact that we can find both the quotient and the remainder functions by dividing \(f(x)\) by \(g(x)\). To do this we need to learn the method for long division of polynomials.

Example 1: Long Division of a Polynomial

In this first example, we see how to divide \(f(x) = 2x^4 - x^3 + 3x^2 + 5x + 4\) by \(g(x) = x^2 -1\).

Now that we've seen the method, let's see how to deal with cases in which one, or more, of the coefficients of \(f(x)\) equals to \(0\). We see this in example 2, in which the \(x^4\) and the \(x\) coefficients are both equal to \(0\).

Example 2: Long Division of a Polynomial

In this second example, we see how to divide \(f(x) = 2x^5 + x^3 - 3x^2 - 8\) by \(g(x) = x^3 +x - 1\).


Exercise 1

For each of the following, find an expression for \(f(x) \div g(x)\) and write an expression for both the quotient function, \(Q(x)\), and the remainder function, \(R(x)\):

  1. \(f(x) = x^4 - 3x^3 + 4x^2 - 4x + 10\) and \(g(x) = x^2 - 3\).
  2. \(f(x) = x^4 - 3x^3 + x^2 - 2x + 3\) and \(g(x) = x -2\).
  3. \(f(x) = 2x^5 + 3x^4 - 6x^2 + x\) and \(g(x) = x^3 + x - 1\).
  4. \(f(x) = x^5 + 4x^3 - 3x + 7\) and \(g(x) = x+1\).
  5. \(f(x) = 4x^5 + 2x^4 - 5x^2 + 3\) and \(g(x) = 2x^2+x - 1\).
  6. \(f(x) = 6x^4 - 3x^3 + x^2 - 2x + 3\) and \(g(x) = 2x-3\).
  7. \(f(x) = 5x^5 - 3\) and \(g(x) = x^2 - 4\).
  8. \(f(x) = 8x^6 + 4x^5 - 3x^2 + 4x + 3\) and \(g(x) = x+1\).

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 1

Solution Without Working

We find the following results:

  1. When dividing \(f(x) = x^4 - 3x^3 + 4x^2 - 4x + 10\) by \(g(x) = x^2 - 3\), we find the following \[{\scriptsize \begin{aligned} \begin{pmatrix} x^4 - 3x^3 + 4x^2 - 4x + 10 \end{pmatrix} \div \begin{pmatrix} x^2 - 3 \end{pmatrix} & = \frac{x^4 - 3x^3 + 4x^2 - 4x + 10}{x^2 - 3} \\ & = x^2 - 3x + 7 + \frac{-13x+31}{x^2 - 3} \end{aligned} }\] Where the quotient function is: \[Q(x) =x^2 - 3x + 7 \] and the remainder fucntion: \[R(x) = -13x+31\]

  2. When dividing \(f(x) = x^4 - 3x^3 + x^2 - 2x + 3\) by \(g(x) = x -2\), we find the following: \[ {\scriptsize \begin{aligned} \begin{pmatrix} x^4 - 3x^3 + x^2 - 2x + 3 \end{pmatrix} \div \begin{pmatrix} x - 2 \end{pmatrix} & = \frac{x^4 - 3x^3 + x^2 - 2x + 3}{x-2} \\ & = x^3-x^2-x-4 - \frac{5}{x-2} \end{aligned} } \] Where the quotient function is: \[Q(x) =x^3-x^2-x-4 \] and the remainder fucntion: \[R(x) = -5\]

  3. When dividing \(f(x) = 2x^5 + 3x^4 - 6x^2 + x\) by \(g(x) = x^3 + x - 1\), we find the following: \[ {\scriptsize \begin{aligned} \begin{pmatrix} 2x^5 + 3x^4 - 6x^2 + x \end{pmatrix} \div \begin{pmatrix} x^3 + x - 1 \end{pmatrix} & = \frac{2x^5 + 3x^4 - 6x^2 + x}{x^3 + x - 1} \\ & = 2x^2 + 3x - 2 + \frac{-7x^2 + 6x - 2}{x^3 + x - 1} \end{aligned} } \] Where the quotient function is: \[Q(x) =2x^2 + 3x - 2 \] and the remainder fucntion: \[R(x) = -7x^2 + 6x - 2\]

  4. When dividing \(f(x) = x^5 + 4x^3 - 3x + 7\) by \(g(x) = x+1\), we find the following: \[ {\scriptsize \begin{aligned} \begin{pmatrix} x^5 + 4x^3 - 3x + 7 \end{pmatrix} \div \begin{pmatrix} x+1 \end{pmatrix} & = \frac{x^5 + 4x^3 - 3x + 7}{x+1} \\ & = x^4 - x^3 + 5x^2 - 5x +2 - \frac{5}{x+1} \end{aligned} } \] Where the quotient function is: \[Q(x) =x^4 - x^3 + 5x^2 - 5x +2 \] and the remainder fucntion: \[R(x) = -5\]

  5. When dividing \(f(x) = 4x^5 + 2x^4 - 5x^2 + 3\) by \(g(x) = 2x^2+x - 1\), we find: \[ {\scriptsize \begin{aligned} \begin{pmatrix}4x^5 + 2x^4 - 5x^2 + 3\end{pmatrix} \div \begin{pmatrix} 2x^2+x - 1 \end{pmatrix} & = \frac{4x^5 + 2x^4 - 5x^2 + 3}{2x^2+x - 1} \\ & = 2x^3 + x - 3 + \frac{4x}{2x^2+x - 1} \end{aligned} } \] Where the quotient function is: \[Q(x) =2x^3 + x - 3 \] and the remainder fucntion: \[R(x) = 4x\]

  6. When dividing \(f(x) = 6x^4 - 3x^3 + x^2 - 2x + 3\) by \(g(x) = 2x-3\), we find: \[ {\scriptsize \begin{aligned} \begin{pmatrix} 6x^4 - 3x^3 + x^2 - 2x + 3 \end{pmatrix} \div \begin{pmatrix} 2x-3 \end{pmatrix} & = \frac{6x^4 - 3x^3 + x^2 - 2x + 3}{2x-3} \\ & = 3x^3+3x^2+5x+\frac{13}{2} + \frac{45}{4x-6} \end{aligned} } \] Where the quotient function is: \[Q(x) = 3x^3+3x^2+5x+\frac{13}{2} \] and the remainder fucntion: \[R(x) = \frac{45}{2}\]

  7. When dividing \(f(x) = 5x^5 - 3\) by \(g(x) = x^2 - 4\), we find: \[ {\scriptsize \begin{aligned} \begin{pmatrix} 5x^5 - 3 \end{pmatrix} \div \begin{pmatrix} x^2 - 4 \end{pmatrix} & = \frac{5x^5 - 3}{x^2 - 4} \\ & = 5x^3 + 20x + \frac{80x-3}{x^2 - 4} \end{aligned} } \] Where the quotient function is: \[Q(x) = 5x^3 + 20x \] and the remainder fucntion: \[R(x) = 80x - 3\]

  8. When dividing \(f(x) = 8x^6 + 4x^5 - 3x^2 + 4x + 3\) by \(g(x) = x+1\), we find the following: \[ {\scriptsize \begin{aligned} \begin{pmatrix} 8x^6 + 4x^5 - 3x^2 + 4x + 3 \end{pmatrix} \div \begin{pmatrix} x+1 \end{pmatrix} & = \frac{8x^6 + 4x^5 - 3x^2 + 4x + 3}{x + 1} \\ & = 8x^5 - 4x^4 + 4x^3 - 4x^2 + x + 3 \end{aligned} } \] Where the quotient function is: \[Q(x) = 8x^5 - 4x^4 + 4x^3 - 4x^2 + x + 3 \] and the remainder fucntion: \[R(x) = 0\]


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