Question

Please help! I'm so confused

Answer 1

The remainder theorem says that the remainder upon dividing a polynomial $p(x)$ by a linear polynomial $x-a$ is the same as the value of $p(x)$ at $x=a$. Dividing by any linear polynomial will always result in the following:

$p(x)=(x-a)q(x)+r(x)$

where $q(x)$ and $r(x)$ are also polynomials. Taking $x=a$, the term involving $q(x)$ vanishes, so that $p(a)=r(a)$ is exactly the remainder upon dividing.

Via synthetic division, we have

... |  2    -9    7    -5    11
4  |         8   -4   12    28
- - - - - - - - - - - - - - - - - -
... |  2    -1    3     7    39

which translates to

$\dfrac{2x^4-9x^3+7x^2-5x+11}{x-4}=2x^3-x^2+3x+7+\dfrac{39}{x-4}$

that is, we're left with a remainder of 39.

Via the remainder theorem, we have

$p(4)=2\times4^4-9\times4^3+7\times4^2-5\times4+11=39$

as expected.