1.9: where and are constants.

4PM1/2R/June/2024 — Question 4 · 11 marks

f(x)=px3+qx2−37x−12q\displaystyle f(x) = px^3 + qx^2 - 37x - 12q where p\displaystyle p and q\displaystyle q are constants.
When f′(x)\displaystyle f'(x) is divided by (x+2)\displaystyle (x + 2) the remainder is −33\displaystyle -33
Given that (x+5)\displaystyle (x + 5) is a factor of f(x)\displaystyle f(x)
(a) (i) show that p=2\displaystyle p = 2
(ii) find the value of q\displaystyle q
(6)
(b) Hence, use algebra to factorise f(x)\displaystyle f(x) completely.
(3)
(c) Hence solve the equation f(x)=0\displaystyle f(x) = 0
(2)