Explore: Synthetic Division with Non-Integers Solutions

Synthetic Division with Non-Integers Solutions

When the divisor has a coefficient other than one , you need to complete an additional step to use synthetic division. 
First, divide every term of the dividend and the divisor by the coefficient of the divisor. 
Then follow the steps of synthetic division.

Example 4

Simplify using synthetic division.

$\frac{18 x^{4} - 15 x^{3} + 30 x^{2} - 21 x + 1}{3 x - 1}$

Note

It is CRITICAL to divide EVERY term by the coefficient of the divisor. If you skip this step, the answer will be off by a multiple of the coefficient.

Implement

$\frac{(18 x^{4} - 15 x^{3} + 30 x^{2} - 21 x + 1) \div 3}{(3 x - 1) \div 3}$

$\frac{6 x^{4} - 5 x^{3} + 10 x^{2} - 7 x + \frac{1}{3}}{x - \frac{1}{3}}$

$\begin{array}{rcl} x - \frac{1}{3} & = & 0 \\ x & = & \frac{1}{3} \end{array}$

Explain

Divide every term by the coefficient of the linear divisor (In this example, the coefficient of 3x is 3.)
Simplify each term
Find the zero of the divisor
Write the remainder using the given divisor (Multiply all terms of the remainder by the number you divided out.)
Write the quotient with the remainder

$\frac{(- 1) (3)}{(x - \frac{1}{3}) (3)} = \frac{- 3}{3 x - 1}$

$6 x^{3} - 3 x^{2} + 9 x - 4 - \frac{3}{3 x - 1}$

Note

In Example 4, if you do not divide each term by the coefficient of the divisor, the error would look like this.

$18 x^{3} - 9 x^{2} + 27 x - 12 - \frac{3}{3 x - 1}$

Example 5

Find the quotient using synthetic division.
$(32 x^{3} + 8 x^{2} - 4 x + 5) \div (4 x + 3)$

$\frac{(32 x^{3} + 8 x^{2} - 4 x + 5) \div 4}{(4 x + 3) \div 4} = \frac{8 x^{3} + 2 x^{2} - x + \frac{5}{4}}{x + \frac{3}{4}} x + \frac{3}{4} = 0 x = - \frac{3}{4}$

$\begin{array}{rcl} (- \frac{1}{4}) (4) & = & - 1 \\ (x + \frac{3}{4}) (4) & = & 4 x + 3 \\ 8 x^{2} - 4 x + 2 - \frac{1}{4 x + 3} \\ or 2 (4 x^{2} - 2 x + 1) - \frac{1}{4 x + 3} \end{array}$