Accompanying Resource: Printable Study Guide
Why Do We Need Synthetic Division?
Synthetic division is often used to find the roots of higher-degree polynomials (degree 3 and up). These roots can be used to factor the polynomial.
Let's say you have a quadratic function (degree 2) and you need to find the roots. The roots of a function are the values that make the function equal to zero. On a graph, they are where the function crosses the x-axis. There are several different ways to find the roots of a quadratic function. You could try factoring it or you could complete the square. There's always the Quadratic Formula too.
But what if you have a function with a higher degree? If you have an x cubed term or an x to the 4th power, you can't use the Quadratic formula. You could try to factor them, but if the answer is an irrational or complex number you're just out of luck.
Thankfully, there is still a way to find the roots or factor a polynomial with a higher degree. It involves dividing the polynomial by a linear factor with a leading coefficient of 1 like x + 4 or x - 3. You could use long division to do this, but synthetic division is a shortcut method that involves only the coefficients of the terms. You get the same answer you would get if you used long division, but most students find that it's much, much faster. It may look complicated at first, but you'll get the hang of it after a few tries.
Review of Long Division
In order to make sense of synthetic division, let's look at the steps involved in long division. Let's say we have the division problem below:
The first thing you would do is rearrange it and put the x - 3 out to the left. If you use long division to solve this, you get:
We see that the answer (in orange) is a quadratic and there is a remainder of 0. Long division can be tricky. You have to be very careful when subtracting negatives and have to make sure you bring down the terms correctly.
Thankfully, synthetic division is a much simpler, faster way to divide polynomials.
Using Synthetic Division
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