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  5. Synthetic Division

Synthetic Division

Synthetic Division is used when dividing polynomials by linear factors.

Steps for Synthetic Division

  1. Setup: Write the coefficients of the polynomial in descending order of their exponents. If there is a missing term, use \( 0 \). Write the divisor \( x - c \) or \( x + c \) equal to \( 0 \), find \( 0 \). If the divisor is \( x - c \), use \( c \). If the divisor is \( x + c\), use \( -c \).
  2. Divide: The initial coefficient, or the coefficient of the phrase with the highest degree, should be moved to the bottom row. Multiply the value of the divisor, \( c \), by the recently brought down number, and record the outcome beneath the subsequent coefficient. Add the new number to the coefficient below it. Until the coefficients run out, keep multiplying and adding till the end.

  3. Result: The remainder is the final number in the bottom row. The coefficients of the quotient are represented by the integers in the second row, omitting the remainder, in descending order of their exponents.

Example

Divide the polynomial \( 2x^3 - 5x^2 + 3x + 1 \) by \( x-2 \) using synthetic division.

First, we can rewrite this expression in the synthetic division format. We can identify the dividend is \( 2x^3 - 5x^2 + 3x + 1 \). We can also identify the divisor is \( x-2 \), thereby making \(c = 2\):

\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 2 & -5 & 3 & 1\\ {\color{red}2} &   &   &   &  \\ \hline &   &   &   & |\phantom{-} { } \end{array}

Next, we can divide the polynomial using synthetic division:

\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 2 & -5 & 3 & 1\\ {\color{red}2} & \downarrow & 4 & -2 & 2\\ \hline & 2 & -1 & 1 & |\phantom{-} {\color{blue}3} \end{array}

Therefore, we can determine that dividing the polynomial \(2x^3 - 5x^2 + 3x + 1\) by \(x-2\) gives us a remainder of \(\boldsymbol{3}\).


Divide the following polynomials using Synthetic Division.

\(3x^4 -2x^3 + 0x^2 + 5x - 6\) by \(x + 1\)

First, we can rewrite this expression in the synthetic division format. We can identify the dividend is \( 3x^4 -2x^3 + 0x^2 + 5x - 6 \). We can also identify the divisor is \( x + 1 \), thereby making \(c = -1\):

\begin{array}{c|rrrr}& x^4 & x^3 & x^2 & x^1 & x^0\\ & 3 & -2 & 0 & 5 & -6\\ {\color{red}{-1}} &   &   &   &   &  \\ \hline &   &   &   &   &|\phantom{-} { } \end{array}

Next, we can divide the polynomial using synthetic division:

\begin{array}{c|rrrr}& x^4 & x^3 & x^2 & x^1 & x^0\\ & 3 & -2 & 0 & 5 & -6\\ {\color{red}{-1}} & \downarrow & -3 & 5 & -5 & 0\\ \hline & 3 & -5 & 5 & 0 &|\phantom{-} {\color{blue}-6} \end{array}

Therefore, we can determine that dividing the polynomial \( 3x^4 -2x^3 + 0x^2 + 5x - 6 \) by \( x + 1 \) gives us a remainder of \(\boldsymbol{-6}\).

\(2x^3 + 7x^2 - 6\) by \(x - 4\)

First, we can rewrite this expression in the synthetic division format. We can identify the dividend is \(2x^3 + 7x^2 - 6\) . We can also identify the divisor is \(x - 4\), thereby making \(c = 4\):

\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 2 & 7 & 0 & -6\\ {\color{red}{4}} &   &   &   &  \\ \hline &   &   &   & |\phantom{-} { } \end{array}

Next, we can divide the polynomial using synthetic division:

\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 2 & 7 & 0 & -6\\ {\color{red}{4}} & \downarrow & 8 & 60 & 60\\ \hline & 2 & 15 & 240 & |\phantom{-} {\color{blue}234} \end{array}

Therefore, we can determine that dividing the polynomial \(2x^3 + 7x^2 - 6\) by \(x - 4\) gives us a remainder of \(\boldsymbol{234}\).

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