Question

What is the remainder when the function `f(x)=x^3-4x^2+12` is divided by (x+2)?

Answer 1

`color(blue)(-12)`

Explanation:

The Remainder theorem states that, when `f(x)` is divided by `(x-a)`

`f(x)=g(x)(x-a)+r`

Where `g(x)` is the quotient and `r` is the remainder.

If for some `x` we can make `g(x)(x-a)=0`, then we have:

`f(a)=r`

From example:

`x^3-4x^2+12=g(x)(x+2)+r`

Let `x=-2`

`:.`

`(-2)^3-4(-2)^2+12=g(x)((-2)+2)+r`

`-12=0+r`

`color(blue)(r=-12)`

This theorem is just based on what we know about numerical division. i.e.

The divisor x the quotient + the remainder = the dividend

`:.`

`6/4=1` + remainder 2.

`4xx1+2=6`

Answer 2

`"remainder "=-12`

Explanation:

`"using the "color(blue)"remainder theorem"`

`"the remainder when "f(x)" is divided by "(x-a)" is "f(a)`

`"here "(x-a)=(x-(-2))rArra=-2`

`f(-2)=(-2)^3-4(-2)^2+12=-12`