- In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
$$f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + b x + 8 a$$
where \(a\) and \(b\) are constants.
Given that ( \(x - 4\) ) is a factor of \(\mathrm { f } ( x )\),
- use the factor theorem to show that
$$10 a = 32 + b$$
Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\),
- express \(\mathrm { f } ( x )\) in the form
$$f ( x ) = ( 2 x + k ) ( x - 4 ) ( x - 2 )$$
where \(k\) is a constant to be found.
- Hence,
- state the number of real roots of the equation \(\mathrm { f } ( x ) = 0\)
- write down the largest root of the equation \(\mathrm { f } \left( \frac { 1 } { 3 } x \right) = 0\)