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Remainder Theorem

Before tackling Remainder theorem, you might want to revisit long division (also known as synthetic division) and quadratic equations. Just like numbers, polynomials can be divided through by numbers.

The Remainder Theorem is a useful mathematical theorem that can be used to factorize polynomials of any degree in a neat and fast manner.

The Remainder Theorem states that when you divide a polynomial P(x) by any factor (x - a); which is not necessarily a factor of the polynomial; you'll obtain a new smaller polynomial and a remainder, and this remainder is the value of P(x) at x = a, i.e P(a)

Remainder Theorem operates on the fact that a polynomial is completely divisible once by its factor to obtain a smaller polynomial and a remainder of zero. This provides an easy way to test whether a value a is a root of the polynomial P(x).

For example, given a polynomial P(x), and also given that a is a root of the polynomial, then when P(x) is divided by the factor (x - a), the result should be a smaller polynomial P1(x) and a remainder zero.

Below is an example that serves to prove the remainder theorem

prove that x = 1 is a root of P(x),

solution:

which implies that x = 1 is a root of the polynomial P(x), and (x - 1) is a factor of P(x)

Therefore if we were to synthetically divide through P(x) by (x - 1), we should get a new smaller polynomial and a remainder of zero:

Remainder Theorem Examples

The first step of syntheic division is to arrange the polynomial and the factor in the format below. The factor is the divisor and is on the outside, while the polynomial is the dividend and goes under the division bar.

The next step is to divide the first term of the polynomial (the first term should be the one with the highest power) by the x part of the factor. In this case, you divide x3 by x, to get x2 which you then write at the top of the division bar.

Next you multiply the term that you wrote at the top by the entire divisor, in this case you multiply x2 by (x - 1) to get (x3 - x2) which you then subtract from the polynomial under the division bar as shown below.

By subtracting, you get rid of the term with the largest exponent to reduce the size of the polynomial which is a remainder as shown below:

Now we have a new polynomial for which we repeat the second step. In this example, we divide -5x2 by x and add the result which is -5xto the term already at the top of the division bar

Next we repeat the third step, we multiply this result which is on top of the division bar (only the most recently added) by the entire divisor and then subtract what you get from the divident polynomial, as shown in this step below:

Once again we have a new polynomial under the division bar.

As you have probably guessed, we repeat the second step by dividing the first term of the polynomial by the x part of the divisor and add the result to whatever is already on top of the division bar.

Next we multiply the most recently added term by the entire divisor and again subtract the result from the dividend polynomial.

Now the subtraction step is repeated to obtain a new polynomial.

The remainder here is zero because we have completely divided through P(x) by the factor (x - 1). When you reach this step, you're done, dividing any further by zero would result in zero.

Now lets try to use the remainder theorem to find a value of P(-1), to see what we result with. We already know that x = -1 is not a root of the polynomial P(x)

The first step is the same as in the previous example; arrange the divisor and dividend on the outside and inside of the division bar respectively.

Next we perform the first division; the first term of the dividend by the x term of the divisor

Then comes the multiplication; the most recent result by the entire divisor and subtract the result from the dividend

The subtraction gets rid of the term with the largest power and yields a new dividend under the division bar.

We once again divide the first term of the dividend by the x term of the divisor and add the result to whatever is already on top of the division bar.

Next we repeat the multiplication step

Then we subtract the result from the dividend to obtain a new polynomial dividend.

Once again we divide the first term of the dividend by the x term of the divisor.

Multiplication once again:

Then subtraction:

and we're left with a remainder. We can't continue division any further since the x term in the divisor has a higher exponent (x1) than the x in the dividend polynomial (which in this case happens to be x0) and we don't want to end up with negative exponents so we stop there and say that whatever is left is the remainder which in this case is -24.

Next we check to see if the value of P(-1) is the same as the remainder obtained above

Thus we have proved the remainder theorem. You may try a few more values of x as a way of practicing synthetic division of polynomials.

Solving Polynomials Using the Remainder Theorem

From the section on polynomials, we know that the root of a polynomial P(x) is defined as the value of x for which the polynomial is equal to zero.

Using Remainder Theorem, we can redefine a root as a value a for which the factor (x - a) divides through the polynomial P(x) to get a remainder of zero. In other words it divides through the polynomial completely.

The remainder theorem is especially useful in finding the roots of polynomials of a large degree (4 or more). By finding the one root and using it to synthetically divide through the polynomial, we're able to obtain a smaller polynomial for which we repeat the process until we find the last root.

The example below better explains the steps involved;

Given the polynomial P(x) below, find all its roots.

Solution:

From the degree of the polynomial P(x) which is 5, we can tell that P(x) will have 5 roots.

The first step is to use small numbers values of x to find which ones will give a value of zero ie we need to find

So we first attempt x = 0, we substitute for x = 0 into P(x)

since P(0) is not equal to zero, we can safely conclude that 0 is not a root of P(x)

Next we move on to another number, let's try substituting for x = -3

P(-3) = 0 which means that -3 is a root of the polynomial and that (x + 3) is a factor of the polynomial. So now we can use synthetic division to get a smaller polynomial

We obtain a new polynomial, lets call this polynomial f(x)

We then repeat the process for f(x). We try to find a root of f(x) by trial and error

First we try x = 1

f(1) is not equal to zero, so we move on to another value of x

Try x = 3

f(3) = 0 which implies that x = 3 is a root of the polynomial and (x-3) is a factor.

Next we divide f(x) by (x - 3)

We obtain a new polynomial g(x) and a remainder of zero

We repeat the process one last time.

Since we can see that g(x) is a polynomial of degree 2 and that there only addition operators in the polynomial, we can conclude the that one of the roots of g(x) must be a negative.

So we try to substitute for x = -4

since g(4) is equal to zero, we conclude that x = -4 is a root of the polynomial and (x+4) is a factor, which allows us to perform synthetic division on g(x)

from which we get the last factor of the P(x) as (x + 3) and thus the root as x = -3.

So now we can rewrite P(x) as

and the roots of P(x) are x = {4,-3,-3,3}

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