The remainder theorem

Remainder theorem: If a polynomial P(x) of degree n ≥ 1 is divided by x - b, where b is a constant, then the remainder is P(b).

A couple of examples illustrating the remainder theorem 

Example #1

In the lesson about polynomial long division, we ended with the following result:

(x2 - 5x + 1) ÷ (x + 3) = x + -8 with a remainder of 25

You can also write (x2 - 5x + 1) = (x + 3)(x + -8) + 25

Observation

(x + 3)(x + -8) + 25 = x2 + -8x + 3x + -24 + 25

(x + 3)(x + -8) + 25 = x2 + -5x + 1

Using (x2 - 5x + 1) = (x + 3)(x + -8) + 25

Dividend = (x2 - 5x + 1)

divisor = (x + 3) 

quotient = (x + -8) 

Remainder = 25  

Dividend = divisor x quotient + remainder

When we divide (x2 - 5x + 1) by (x + 3), we get a remainder of 25.

Using P(x) = (x2 - 5x + 1), calculate P(-3).

P(-3) = (-3)2 - 5(-3) + 1

P(-3) = 9 - -15 + 1

P(-3) = 9 + 15 + 1

P(-3) = 25

As you can see, if a = -3, P(-3) is equal to the remainder (25) when (x2 - 5x + 1) is divided by (x + 3)

Example #2

In the lesson about polynomial long division, we ended also with the following result:

(x2 + 3x - 10) ÷ (x - 2) = x + 5

(x2 + 3x - 10) = (x - 2)(x + 5) 

(x2 + 3x - 10) = (x - 2)(x + 5) + 0

P(2) = 22 + 3(2) - 10

P(2) = 4 + 6 - 10

P(2) = 10 - 10

P(2) = 0

Again, you can see that if a = 3, P(2) is equal to the remainder (0) when (x2 - 5x + 1) is divided by (x - 2)

Recent Articles

  1. How To Find The Factors Of 20: A Simple Way

    Sep 17, 23 09:46 AM

    Positive factors of 20
    There are many ways to find the factors of 20. A simple way is to...

    Read More

  2. The SAT Math Test: How To Be Prepared To Face It And Survive

    Jun 09, 23 12:04 PM

    SAT math
    The SAT Math section is known for being difficult. But it doesn’t have to be. Learn how to be prepared and complete the section with confidence here.

    Read More

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Math quizzes

 Recommended

Math vocabulary quizzes