Remainder Theorem: Definition & Examples

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  • 0:04 Remainder Theorem Definition
  • 0:57 Remainder Theorem Function
  • 1:59 How to Use the Theorem
  • 3:51 Some Example Problems
  • 5:58 Lesson Summary
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Lesson Transcript
Instructor: Joshua White

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

This lesson will examine the remainder theorem and how to use it to evaluate a function. It will additionally show how the remainder theorem can be used to find the zero(s) of a function.

Defining Remainder Theorem

Everyone loves to find a shortcut whether it involves driving directions or some other type of long task. Discovering a quicker and more efficient way to arrive at the same end point makes you feel good since you've most likely saved time, effort, and/or money. Math is filled with these types of shortcuts and one of the more useful ones is the remainder theorem.

The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x - a, the remainder of that division will be equivalent to f(a). In other words, if you want to evaluate the function f(x) for a given number, a, you can divide that function by x - a and your remainder will be equal to f(a).

It should be noted that the remainder theorem only works when a function is divided by a linear polynomial, which is of the form x + number or x - number. How does the remainder theorem save you time? Let's find out.

Remainder Theorem Function

The remainder theorem is especially useful when it is paired with synthetic division. If you remember, synthetic division is an alternate method to quickly and easily divide polynomials instead of using long division. Also, remember that in synthetic division, the number in the bottom row in the last column on the right is the remainder. Thus, rather than plugging a value in and using order of operations, you can use synthetic division as a way to evaluate a polynomial for a given value.

Additionally, synthetic division and the remainder theorem can be used to determine if a value is a zero of a function. Hopefully, you remember that a zero of a function, by definition, is any point c, where f(c) = 0. Therefore, if you find a remainder of zero after performing synthetic division, the number listed out front, referred to as a in the definition above, evaluates to zero, or f(a) = 0.

Note, that you can use long division instead of synthetic division, but it's almost always faster and easier to use synthetic division.

How to Use the Theorem

Let's take the function f(x) = x^4 + 3x^3 - 6x^2 - 6x + 8. Suppose you were told to evaluate it for x = 3. You could spend the time to plug in a 3 for every x (all four of them) listed above. Then you could perform the order of operations (PEMDAS anyone?) to evaluate all five terms. Finally, you would combine all the like terms to get the final answer for f(3).

But it's so much quicker and easier to use the remainder theorem. Simply setup synthetic division where you will divide x^4 + 3x^3 - 6x^2 - 6x + 8 by x - 3, and you'll have you answer in no time. Remember, that you will divide by x - 3 (not x + 3) because a = 3 in this example and the remainder theorem is based on dividing by x - a (not x + a). You should get 98 as the remainder, which means that f(3) = 98. The work is shown below.

Remainder Theorem Example 1

Now, let's evaluate the same function for x = -4. You can use the same process of dividing x^4 + 3x^3 - 6x^2 - 6x + 8 by x + 4. Note, that in this example, since a = -4, x - a will be x - (-4), or x + 4.

After you perform the synthetic division, you should notice several things. The first is that the remainder is 0; therefore, the remainder theorem tells you that f(-4) = 0. However, since the remainder is 0 (and not any other number), there is something else you should have discovered. You've also determined that -4 is a zero of f(x) since, as mentioned above, f(-4) = 0. The full work is shown below.

Remainder Theorem Example 2

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