Question

4. The term containing the highest power of x in the polynomial f(x) is 4x^4. Given that f(x) = 0

has 2 roots -2 and 3, and that 2x² + 3x + 2 is a quadratic factor of f(x),
(a) express f(x) as a polynomial in descending powers of x,

Answer 1

We're given
`f(x) = 0` when
`x=-2` and
`x=3`, so both
`x+2` and
`x-3` divide
`f(x)`

We're also told
`2x^2+3x+2` divides
`f(x)`. This quadratic does not have roots at -2 or 3, so we can factorize
`f(x)` as

`f(x) = 2 (x + 2) (x - 3) (2x^2 + 3x + 2)`

where the leading coefficient is 2 because the full expansion should have a leading term of
`4x^4`.

Now just expand
`f(x)` :

`f(x) = 2 (x + 2) (x - 3) (2x^2 + 3x + 2)`

`f(x) = 2 (x^2 - x - 6) (2x^2 + 3x + 2)`

`f(x) = 2 (2x^4 + x^3 - 13x^2 - 20x - 12)`

`f(x) = \boxed{4x^4 + 2x^3 - 26x^2 - 40x - 24}`