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)

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