Question

How do you use the remainder theorem to find P(c) `P(x)=6x^3-x^2+4x+3`,a=3?

Answer 1

`P(3)=168`

Explanation:

According to remainder theorem when we divide a polynomial `P(x)` (here `P(x)=6x^3-x^2+4x+3`), by a binomial of degree one say `(x-a)`, (here `(x-3)` as we desire `P(a)` wih `a=3`)

the remainder is `P(a)`

Normally to find `P(a)`, one substitute `x` with `a`, but as we have to use Remainder theorem, we should divide `P(x)` by `(x-a)` and then the remainder would be `P(a)`. For this let us use synthetic division to divide `6x^3-x^2+4x+3` by `(x-3)`.

`3|color(white)(X)6" "color(white)(X)-1color(white)(XXX)4" "" "3`
`color(white)(1)|" "color(white)(XXxx)18color(white)(XXX)51color(white)(XX)165`
`" "stackrel("—————————————)`
`color(white)(x)|color(white)(X)color(blue)6color(white)(Xxxx)color(blue)17color(white)(XXX)color(blue)55color(white)(XX)color(red)168`

i.e. while quotient is `6x^2+17x+55`, remainder is `168`.