Synthetic Division of Polynomials

(Dividing Polynomials by Linear Functions)


In this section we learn about synthetic division of polynomials. This will provide us with a quick method for dividing polynomials by linear functions using the nested scheme, a.k.a Horner's Method.

For instance, by the end of this section we'll know how to quickly find the quotient and remainder functions for the following division: \[\begin{pmatrix} 3x^5 - 2x^3 + x^2 - 3x + 10\end{pmatrix} \div \begin{pmatrix} x - 2 \end{pmatrix} \]

as well as divisions such as:
\[\begin{pmatrix} 4x^3 + x^2 - 5x + 3\end{pmatrix} \div \begin{pmatrix} 2x + 1 \end{pmatrix} \]

These two introductory examples show the different scenarious we learn here:

  • division by linear functions of the type \(x + c\)
  • division by linear functions of the type \(ax + b\) (where the \(x\) coefficient isn't \(1\))
We learn the method for each of these in the tutorials below.

Tutorial 1 : dividing polynomials by \(x+c\)

In this first tutorial we see two examples. We divide divide \(f(x) = x^5 + 2x^3 - 3x^2 - 4x + 5\) by \(g(x) = x-2\) as well as \(f(x) = 4x^4 - 2x^3 + 7x + 10\) by \(g(x) = x+1\).


Exercise 1

  1. Given \(f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7\) and \(g(x) = x-1\), find: \[f(x)\div g(x)\] and clearly state the follow of both the quotient function, \(Q(x)\) and the remainder \(R\).

  2. Using synthetic division, find: \[\frac{2x^5 + x^4 - 2x^3 + 3x^2 - 3x + 16}{x+2}\] and clearly state the follow of both the quotient function, \(Q(x)\) and the remainder \(R\).

  3. Using synthetic division, find: \[\begin{pmatrix}x^6 - x^4 + 2x^3 - 3x^2 - 25x - 100\end{pmatrix} \div \begin{pmatrix}x - 3 \end{pmatrix}\] and clearly state the follow of both the quotient function, \(Q(x)\) and the remainder \(R\).

  4. Given \(f(x) = -2x^5 + 3x^3 - x^2 + 4x + 10\) and \(g(x) = x - 2\), find \(f(x) \div g(x)\) and clearly state the follow of both the quotient function, \(Q(x)\) and the remainder \(R\).

  5. Using synthetic division, find: \[\begin{pmatrix}4x^4+2x^3 - 7x + 5 \end{pmatrix} \div \begin{pmatrix} x - \frac{1}{2} \end{pmatrix} \] and clearly state the follow of both the quotient function, \(Q(x)\) and the remainder \(R\).

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

Solution Without Working

  1. Given \(f(x) = x^4 - 3x^3 + 2x^2 - 5x + 7\) and \(g(x) = x-1\), we find: \[f(x)\div g(x) = x^3 - 2x^2 - 5 + \frac{2}{x - 1}\] Where the quotient function is: \[Q(x) = x^3 - 2x^2 - 5\] and the remainder is: \[R = 2\]

  2. We find: \[\frac{2x^5 + x^4 - 2x^3 + 3x^2 - 3x + 16}{x+2} = 2x^4 - 3x^3 + 4x^2 - 5x + 7 + \frac{2}{x+2}\] Where the quotient function is: \[Q(x) = 2x^4 - 3x^3 + 4x^2 - 5x + 7\] and the remainder is: \[R = 2\]

  3. We find: \[\begin{pmatrix}x^6 - x^4 + 2x^3 - 3x^2 - 25x - 100\end{pmatrix} \div \begin{pmatrix}x - 3 \end{pmatrix} = x^5 + 3x^4 + 8x^3 + 26x^2 + 76x + 200 + \frac{500}{x - 3}\] Where the quotient function is: \[Q(x) = x^5 + 3x^4 + 8x^3 + 26x^2 + 76x + 200 \] and the remainder is: \[R = 500\]

  4. Given \(f(x) = -2x^5 + 3x^3 - x^2 + 4x + 10\) and \(g(x) = x - 2\), we find: \[\begin{aligned} f(x) \div g(x) & = -2x^4 - 4x^3 - 5x^2 -11x - 18 + \frac{-26}{x-2} \\ & = -2x^4 - 4x^3 - 5x^2 -11x - 18 - \frac{26}{x-2} \end{aligned}\] Where the quotient function is: \[Q(x) = -2x^4 - 4x^3 - 5x^2 -11x - 18\] and the remainder is: \[R = -26\]

  5. We find: \[\begin{aligned} \begin{pmatrix}4x^4+2x^3 - 7x + 5 \end{pmatrix}\div \begin{pmatrix} x - \frac{1}{2} \end{pmatrix} & = \frac{4x^4+2x^3 - 7x + 5}{x - \frac{1}{2}} \\ & = 4x^3 + 4x^2 + 2x - 6 + \frac{2}{x- \frac{1}{2}} \end{aligned}\] Where the quotient function is: \[Q(x) = 4x^3 + 4x^2 + 2x - 6\] and the remainder is: \[R = 2\]

Tutorial 2 : dividing polynomials by \(ax+b\)

In this second tutorial we learn how to divide a polynomial by a linear function whose \(x\) coefficient isn't equal to \(1\). In particular we divide \(f(x) = 2x^4 - 4x^2 - 6x - 5\) by \(g(x) = 2x - 6\).


Exercise 2

For each of the following, use syntehtic division to find \(f(x) \div g(x)\). In each case, specify the quotient polynomial function, \(Q(x)\), as well as the remainder \(R\).

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

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

  3. \(f(x) = 9x^3 + 3x^2 - 12x + 7\) and \(g(x) = 3x - 2\).

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

  5. \(f(x) = x^4 - 3x^2 + 4x + 3\) and \(g(x) = \frac{x}{2} - 2\).

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 2

Solution Without Working

  1. We find: \[f(x) \div g(x) = 3x^2 - 4x + 4 - \frac{11}{2x+4}\] Where the quotient polynomial is: \[Q(x) = 3x^2 - 4x + 4\] and the remainder is: \[R = -11\]

  2. We find: \[f(x) \div g(x) = 6x^3+ 3x^2 - 2x + 1 + \frac{8}{2x-1}\] Where the quotient polynomial is: \[Q(x) = 6x^3 + 3x^2 - 2x + 1\] and the remainder is: \[R = 8\]

  3. We find: \[f(x) \div g(x) = 3x^2 + 3x - 2 + \frac{3}{3x-2}\] Where the quotient polynomial is: \[Q(x) = 3x^2 + 3x - 2\] and the remainder is: \[R = 3\]

  4. We find: \[f(x) \div g(x) = \frac{x^4}{3} - \frac{2}{3}x^3 + \frac{5}{3}x^2 - 2x + \frac{5}{3} - \frac{4}{3x+6}\] Where the quotient polynomial is: \[Q(x) = \frac{x^4}{3} - \frac{2}{3}x^3 + \frac{5}{3}x^2 - 2x + \frac{5}{3}\] and the remainder is: \[R = -4\]

  5. We find: \[f(x) \div g(x) = 2x^3 + 4x^2 + 2x + 12 + \frac{15}{\frac{x}{2} - 1}\] Where the quotient polynomial is: \[Q(x) = 2x^3 + 4x^2 + 2x + 12\] and the remainder is: \[R = 15\]


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