Question

How do you synthetic division and the Remainder Theorem to find P(–3) if `P(x) = x^4 + 19x^3 + 108x^2 + 236x + 176`?

Answer 1

The remainder is `8` and the quotient is `=x^3+16x^2+60x+56`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `-3` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `19` `color(white)(aaaaaa)` `108` `color(white)(aaaa)` `236` `color(white)(aaaaa)` `176`

`color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaa)` `-3` `color(white)(aaaaa)` `-48` `color(white)(aaa)` `-180` `color(white)(aaa)` `-168`

`color(white)(aaaaaaaaa)` #`_________________________________________________________`#

`color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `16` `color(white)(aaaaaa)` `60` `color(white)(aaaaaa)` `56` `color(white)(aaaaaa)` `color(red)(8)`

The remainder is `8` and the quotient is `=x^3+16x^2+60x+56`

Apply the remainder theorem for verification

When a polynomial `f(x)` is divided by `(x-c)`, we get

`f(x)=(x-c)q(x)+r`

Let `x=c`

Then,

`f(c)=0+r`

Here,

`f(x)=x^4+19x^3+108x^2+236x+176`

Therefore,

`f(-3)=(-3)^4+19*(-3)^3+108(-3)^2+236*(-3)+176`

`=8`

The remainder is `=8`