Question

How do you use the remainder theorem to find the remainder for each division `(4x^3+4x^2+2x+3)div(x-1)`?

Answer 1

There are a few ways we can find the remainder. The two most common are long division and synthetic division. I prefer synthetic division, so I'll be using that.

`1|4color(white)(..)4color(white)(..)2color(white)(..)3`
`color(white)(1)|color(white)(....4)4color(white)(..)8color(white)(..)10`
`color(white)(1|)color(black)(------)`
`color(white)(1)color(white)(|)4color(white)(..)8color(white)(..)10color(white)(.)13`

That leaves us with `4x^2+8x+10` and a remainder of `13`. To write this as an equation, we need to that our remainder and place it over the divisor (`x-1`). That means our final solution is `y=4x^2+8x+10+13/(x-1)`.

Answer 2

`x-1` is not a factor. The remainder will be 13.

Explanation:

The remainder theorem is a quick and useful way to determine whether an expression as a factor before you actually go ahead with the whole dividing process.

Let: `f(color(blue)(x)) = 4x^3+4x^2 +2x+3`

The divisor is `(x-1)` . Set it equal to 0 and solve for `x`
`x-1 =0 " "rarr color(blue)(x =1)`

`fcolor(blue)((1)) = 4color(blue)((1))^3+4color(blue)((1))^2 +2color(blue)((1))+3 = 13`

If f(x) =0, then the expression is a factor.
Here, `f(1) = 13, so (x-1)` is NOT a factor, and the remainder will be `13`