Question

Use synthetic division to find P(-3) for P(x)=x^4-4x+4


Please show your work

Answer 1

By the polynomial remainder theorem, the remainder of
`P(x)` upon dividing by
`x+3` is equal to the value of
`P(-3)`.

Synthetic division yields

-3 | 1 0 0 -4 4

. | -3 9 -27 93

- - - - - - - - - - - - - - - - - - -

. | 1 -3 9 -31 97

which translates to

`(x^4-4x+4)/(x+3)=x^3-3x^2+9x-31+(97)/(x+3)`

`\implies x^4-4x+4=(x+3)(x^3-3x^2+9x-31)+97`

Then when
`x=-3`, the first term on the right side vanishes and we have
`P(-3)=97`.