$$f(x)=2x^{3}-3x^{2}-11x+6$$
(a) Use the factor theorem to show that $(x-3)$ is a factor of $f(x)$. [2]
(b) Hence fully factorise $f(x)$. [3]
(c) Hence solve the equation $$2(8^{y})-3(4^{y})-11(2^{y})+6=0$$ giving your answers to 3 significant figures where appropriate. [3]
- (a) for attempting $f(3)$M1
- (a) $f(3)=54-27-33+6=0$ and conclusion so $(x-3)$ is a factorA1
- (b) for dividing by $(x-3)$ or comparing coefficients to obtain a quadratic factor $2x^{2}+\dots x\pm 2$M1
- (b) $(x-3)(2x^{2}+3x-2)$A1
- (b) $(x-3)(2x-1)(x+2)$A1
- (c) for recognising $x=2^{y}$ and solving $2^{y}=$ one of their positive roots using logsM1
- (c) $y=-1$A1
- (c) $y=\log_{2}3=1.58$ (awrt), with $2^{y}=-2$ rejected or ignoredA1
- Answer: (a) $f(3)=0$ (b) $(x-3)(2x-1)(x+2)$ (c) $y=-1,\ y=1.58$
- Examiner tip: In (c) the link is $8^{y}=(2^{y})^{3}$; reject $2^{y}=-2$ because $2^{y}>0$ – don't try to take $\log$ of a negative.