Remainder Theorem and Polynomials

The polynomial remainder theorem states that when any polynomial p(x) with a degree of one or a greater number is divided by (x - a), a linear polynomial where a is any real number, you obtain p(a) as a remainder. 

When it comes to the Euclidean division, the division of real numbers is fairly simple. You take a number, say 24, divide it by 5. You get a quotient of 4 and a remainder of 4. Thus, you can conclude that 24 = (5 x 4) + 4. If you divide the same 24 by 4, you get a quotient of 6 and a remainder of 0. So both 4 and 6 are factors of 24, and 24 is a multiple of both 4 and 6.

But when it comes to the Euclidean division of polynomials, things can get long and complicated. The remainder theorem and Factor theory are concepts that make this easier for you.

Understanding Remainder Theorem

Polynomial remainder theorem, otherwise known as little Bezout’s theorem gives us a method of identifying the remainder of a polynomial divided by a linear equation. If we divide a polynomial p(x) with a linear equation (x-a), the resulting remainder would be p(a). If the p(a) is 0, it means that the linear equation (x-a) is a factor of the polynomial p(x) (this is called factor theorem).

Let’s take a polynomial equation p(x) = x² + 6x - 3, when you divide with a linear polynomial x-3, the remainder should be p(3).

p(3) = (3)² + 6(3) - 3

       = 9 + 18 - 3

       = 24

The remainder is now 24.

Long Divison Verification

Let’s verify this now with the traditional long division method:


x-3 \[\sqrt{x^{2} + 6x - 3}\]

         - x² - 3x

                  9x  - 3

               -  9x  - 27

                        = 24

So here, we have our p(x) = x² + 6x - 3 divided by x - 3 in the long division method giving us a quotient of x+9 and a remainder 24.

Thus we can verify that p(x) = x² + 6x - 3 divided by (x - 3) will give us a reminder p(3).

You can verify this with other polynomials too. Before you divide a polynomial with a non-zero linear equation, make sure that:

  • The terms of a polynomial are arranged in the descending order of their degrees.

  • You divide the first term of the polynomial dividend with your divisor’s first term to obtain your first quotient. 

  • The resulting terms after the subtraction act as your next dividend while the divisor remains the same.

  • Continue the process until your dividend has a lesser degree than that of your divisor.

Factor Theorem

As we hinted earlier, the factor theorem is basically the inverse of the polynomial remainder theorem. As we discussed earlier when you divide 24 with 4 you get a remainder of 0, thus concluding 4 being a factor of 24. Similarly, if you divide a polynomial p(x) with a linear equation (x-a) and get the remainder as zero, it means that the linear equation x-a is a factor of the polynomial.

Remainder Theorem Formula and Proof

So from our understandings so far, we can identify that a remainder theorem equation would be:

p(x) = (x-a) * q(x) + r(x)

Where r(x) equals p(a).

Now let us prove this. Since r(x) is a constant, it can just be r. Now for our p(a)

p(a) = (a-a) * q(a) + r

       = 0 * q(a) + r

       = r

Thus we get the result p(a) = r, the remainder.

Remainder Theorem Examples

Let us now take a look at a couple of remainder theorem examples with answers.

Example 1:

What would be the remainder when you divide x³+4x²-2x + 5 by x-5?


p(x)=  x³+4x²-2x+5

Divisor = x-5 

p(5) = (5)³ + 4 (5)² - 2 (5) +5 = 125 + 100 - 10 + 5 = 220

Example 2:

What would be the remainder when you divide 3x²+15x-45 by x-15?


p(x) = 3x²+15x-45

Divisor = x-15

p(15) = 3 (15)² + 15 (15) - 45 = 675 + 225 - 45 = 855

In this way, the remainder theorem has made it easy for us all to find the remainders of polynomial equations divided by linear equations without having to resort to the more complex long division method.