Find the remainder when x² + 3x – 11 is divided by x – 2.

The Remainder Theorem says the remainder will equal P(a). Here a is 2.

P(2) = (2)² + 3(2) – 11

= 4 + 6 – 11

= -1

Our remainder is -1.

Find the remainder when 3x² – 2x + 2 is divided by x + 1.

Here a is -1, not +1. The remainder when dividing the polynomial by x + 1 is:

P(-1) = 3(-1)² – 2(-1) + 2

= 3 + 2 + 2

= 7

Find the remainder when 3x³ + x² – 4x – 1 is divided by x – 2.

The Remainder Theorem works for polynomials of any degree in the numerator, but it can only divide by 1st degree polynomials in the denominator.

Plugging in 2 gets us:

P(2) = 3(2)³ + (2)² – 4(2) – 1

= 24 + 4 – 8 – 1

= 19

The remainder is 19, perhaps our least favorite prime. It "borrowed" our jacket one time without asking, and now it smells weird and it won't wash out. You're gross, 19.

Find the remainder when x⁴ – 2x³ – x² + 2x – 2 is divided by x – 4.

Plug x = 4 into our polynomial to find the remainder.

P(4) = (4)⁴ – 2(4)³ – (4)² + 2(4) – 2

= 256 – 128 – 16 + 8 – 2

= 118

That is not even close, 4. Not. Even. Close.