Question

Using synthetic division and the remainder theorem, what is `P(a)`if `P(x) = x^3 + 5x^2 - 2x +3`; a = 3?

Answer 1

The answer is `P(x)=(x-3)(x^2+8x+22)+69`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `3` `color(white)(aaaaa)` `|` `color(white)(aaaa)` `1` `color(white)(aaaa)` `5` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `3`
`color(white)(aaaaaaaaaaaa)`________

`color(white)(aaaa)` ###color(white)(aaaaaaa)#`|` `color(white)(aaaa)` ###color(white)(aaaa)#`3` `color(white)(aaaaa)` `24` `color(white)(aaa)` `66`
`color(white)(aaaaaaaaaaaa)`________
`color(white)(aaaa)` ###color(white)(aaaaaaa)#`|` `color(white)(aaaa)` `1` `color(white)(aaa)` `8` `color(white)(aaaaa)` `22` `color(white)(aaa)` `color(red)(69)`

`P(x)=(x-3)(x^2+8x+22)+69`

The remainder is `=69` and the quotient is `=x^2+8x+22`

The remainder theorem is :

When we divide the polynomial `f(x)` by `(x-c)`

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

`f(c)=0+r`

Applying the remainder theorem

`P(3)=(3^3)+(5*3^2)-(2*3)+3`

`=27+45-6+3`

`=69`

The remainder is `=69`