Remainder Theorem

How to use the remainder theorem to find the remainder of a rational expression: theorem, examples, and their solutions.

Theorem

The remainder of f(x) ÷ (x - a) is
f(a).
This is the remainder theorem.

f(x) = (x - a)⋅(quotient) + (remainder)

So f(x) = (x - a)⋅(quotient) + f(a).

Example 1: Remainder of (x³ - 7x + 11)/(x - 2)

Set f(x) = x³ - 7x + 11.

Then the given expression is
f(x)/(x - 2).

By the remainder theorem,
the remainder of f(x)/(x - 2) is
f(2).

So (remainder) = f(2).

Put 2 into f(x).

Then f(2) = 2³ - 7⋅2 + 11.

2³ = 8
-7⋅2 = -14

-14 + 11 = -3

8 - 3 = 5

So the remainder of the given expression is 5.

Example 2: Remainder of (2x⁴ + x³ - 5x² + 3x + 4) ÷ (x + 1)

Set f(x) = 2x⁴ + x³ - 5x² + 3x + 4.

Then the given expression is
f(x)/(x - (-1)).

By the remainder theorem,
the remainder of f(x)/(x - (-1)) is
f(-1).

So (remainder) = f(-1).

Put -1 into f(x).

Then f(-1) = 2⋅(-1)⁴ + (-1)³ - 5⋅(-1)² + 3⋅(-1) + 4.

2⋅(-1)⁴ = 2⋅1
+(-1)³ = -1
-5⋅(-1)² = -5⋅1
+3⋅(-1) = -3

2⋅1 = 2
-5⋅1 = -5

Cancel -1 and +1.
(dark gray terms)

Then the right side is, 2 - 5, -3.

So the remainder of the given expression is -3.

Example 3: (x⁵ - 8x⁴ + 7x² - 3x + a) ÷ (x - 1), Remainder 11, a = ?

Set f(x) = x⁵ - 8x⁴ + 7x² - 3x + a.

Then the given expression is
f(x) ÷ (x - 1).

By the remainder theorem,
the remainder of f(x) ÷ (x - 1) is
f(1).

So (remainder) = f(1).

Put 1 into f(x).

Then f(1) = 1⁵ - 8⋅1⁴ + 7⋅1² - 3⋅1 + a.

1⁵ = 1
-8⋅1⁴ = -8
+7⋅1² = +7
-3⋅1 = -3

1 - 8 = -7

Cancel -7 and +7.
(dark gray terms)

Then (right side) = -3 + a.

It says
the remainder of the given expression is 11.

So the right side is also 11.

So set -3 + a = 11
and find the value of a.

Move -3 to the right side.

Then a = 14.

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