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


`(2x^4-9x^3+7x^2-5x+11)/(x-4)=2x^3-x^2+3x+7+(39)/(x-4)`

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

Via the remainder theorem, we have


`p(4)=2*4^4-9*4^3+7*4^2-5*4+11=39`

as expected.