# Synthetic Division with a Quadratic Divisor

In class and in the book, it is said that synthetic division only works if the divisor is a linear function, i.e., something that can be written in the form (x + k). That’s not true, but synthetic division with higher-order polynomials is a little more complicated.

Here’s an example of synthetic division using a linear divisor. Consider $$(3x^3 + 2x^2 – x + 7)\div (x – 2)$$. Here it is using synthetic division: $\begin{array}{cccccc}2&|&3&2&-1&7\\&|&\downarrow&6&16&30\\&&——&——&——&——\\&&3&8&15&37\end{array}$

The quotient is $$3x^2 + 8 + 15 + \frac{37}{x – 2}$$.

Synthetic division for higher order divisors relies on the same concept, but needs a new line for each coefficient. For instance, a quadratic divisor uses two lines instead of one; a cubic divisor uses three lines; and so on.

For instance, $$(x^2 + 3x – 7)(x^2 – 5x + 2) = x^4 – 2x^3 -20x^2 + 41x – 14$$. To get a remainder, we’ll change the last two terms. To the left, we’ll reverse the signs of both of the coefficients of the lower terms of the divisor; as before, we’ll write all of the coefficients of the dividend: $\begin{array}{cccccccc}-3&7&|&1&-2&-20&40&-10\\ & &|&\downarrow&-3&15&-6& \\ & &|&\downarrow&&7&-35&14\\ & & &——&——&——&——&——\\ & & &1&-5&2&-1&4\end{array}$
In the second row, we use opposite of the coefficient from the $$x$$ term (that is, -3) as a multiplier; in the third row, we use opposite of the constant (that is, 7).

This gives us a result of $$x^2 – 5x + 2 + \frac{-x + 4}{x^2 + 3x – 7}$$, which is the same thing we get from long division.

Notice that, regardless, the lead coefficient ($$a$$) of the divisor must be 1. You can use synthetic division with, say, a divisor of $$4x – 5$$, but you’d have to use $$x – 5/4$$ instead, and then divide each coefficient of the result by 4.

For example, consider $$(2x^3 + x^2 – 17x + 14)/(2x + 7)$$. This is what synthetic division yields: $\begin{array}{cccccc}-7/2&|&2&1&-17&14\\&|&\downarrow&-7&21&-14\\&&——&——&——&——\\&&2&-6&4&0\end{array}$

The actual quotient is $$x^2 – 3x + 2$$, with no remainder, which is what we get from dividing each of the resulting values by 2.