What is the Remainder Theorem? (With Examples)

We are always looking for tools that can help make our math journey just a little bit easier. The remainder theorem is a tool that most people forget about or don’t consider how useful this theorem is.

The remainder theorem states that if we take a polynomial, f(x), and divide it by a binomial, (x-c), the remainder will result in f(c).

Let’s look into how this works and how it can make your life just a bit easier, whether you are taking algebra or calculus or any math in between.

Let’s break down the remainder theorem.

In basic math, a remainder is what you get when you divide a number into another number, but it doesn’t divide evenly. The bit that is left over is a remainder. For instance, 9 / 2 = 4 with a remainder of 1. If we reverse the process, we can multiply 4 * 2 and add in 1 to get back to 9.

When it comes to polynomials, the remainder theorem is almost the same, but it provides more information to us. We divide our polynomial by a binomial, but the remainder that we get tells us what our polynomial is equal to when we place that value back into the polynomial.

Evaluating a simple polynomial at a value of x=c isn’t that bad. But when dealing with complex polynomials, the calculations can be intense. The remainder theorem is a way to evaluate the polynomial without all the hassle.

Dividing a polynomial by a binomial can either be done through long division or synthetic division. Both methods will provide the same result.

There is something small that you should remember. At x = c, the binomial form changes, depending on whether the value is positive or negative.

Also, remember that in long division, we use the binomial form of x=c, which is either x-c or x+c. In synthetic division, we use the value of c.

If we have the polynomial, f(x) = x4+8x3+17x2-2x-24, and we are trying to calculate the value at x=-2, we could plug in -2 and calculate it out. If this polynomial were simpler, that would be an easy task. And with a calculator, the calculation would not be too bad.

But let’s show you the magic of using the remainder theorem.

I am going to use long division, synthetic division, and plugging in the value to show that all three ways will get us to the goal of finding f(c)

I know that all of this sounds confusing but stay with me.

How to use the remainder theorem using long division.

We take the binomial and divide it into our polynomial with long division to find the value f(c).

The value that we are plugging into the polynomial is x=-2. That means that our binomial is going to be x+2.

Long division looks like regular long division, but we are using polynomials. If this concept is confusing, I have another post breaking down this concept so that you can better understand it. At the end of this section, a short video will show the steps I took using long division.

The value that we are testing is x=-2. Our binomial will be represented by x+2, as shown below. It will be the divisor in our long division problem.

We then start our calculations. We find out how many times x+2 divides into x4 and then put the result just below our polynomial. Once we subtract our result, we can bring the next value down, 17x2. We continue the process until no values remain.

The value at the end is the remainder. This value represents what our polynomial would equal if we plugged in the value of x=-2. So, in this case, f(-2)=0.

I have included a short video breaking down the process to make things clearer. It takes the process I used above and breaks it down step by step.

How to use the remainder theorem using synthetic division.

This next section looks at how the remainder theorem and synthetic division make solving certain polynomials a bit easier.

The first step is to set up the polynomial for synthetic division. You will take each coefficient of your polynomial, starting from the highest and going down to the lowest, and you will list them out. The divisor will be the value of x that you are testing.

If there are any missing degrees in your polynomial, we will list them as a zero in the list.

Once we get everything listed, we add and multiply until the final digit. That final digit is our remainder. If the value is zero, that value of x is a root factor or a zero of that polynomial. It zeros out the polynomial when placed back in and evaluate our polynomial at x=c.

If there is an actual value in the last place, this value represents the remainder. The remainder, in this case, is the value of f(c).

Here is another short video that combines the remainder theorem with synthetic division.

Plugging in the value c into our polynomial.

I wanted to show that you should get the same result regardless of your chosen method. This final section shows what happens when we plug -2 back into the polynomial. Our results end up being the same as the ones that we got before.

Example problems.

I wanted to include a few example problems to see how this works out. I broke each problem down, step by step.

Please feel free to play and pause when you need to.

Great job! Keep practicing.


Just a nerd who loves math. Trying to help the world one problem at a time.

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