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

$\dfrac{x^4-4x+4}{x+3}=x^3-3x^2+9x-31+\dfrac{97}{x+3}$

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

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