7.
$$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + b x - 10 \text {, where } a \text { and } b \text { are constants. }$$
Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\),
- use the factor theorem to show that \(2 a + b = - 7\)
Given also that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 36
- find the value of \(a\) and the value of \(b\).
\(\mathrm { f } ( x )\) can be written in the form
$$\mathrm { f } ( x ) = ( x - 2 ) \mathrm { Q } ( x ) \text {, where } \mathrm { Q } ( x ) \text { is a quadratic function. }$$ - Find \(\mathrm { Q } ( x )\).
- Prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.
You must justify your answer and show all your working.