1.10: Past-paper question 5

4PM1/2R/June/2025 — Question 5 · 8 marks

f(x)=x3+x2+x+cwherec is a constantf(x) = x^3 + x^2 + x + c \quad \mathrm{where} c \text{ is a constant}
The remainder when f(x)\displaystyle f(x) is divided by (x−2)\displaystyle (x - 2) is 4 times the remainder when
f(x)\displaystyle f(x) is divided by (x+1)\displaystyle (x + 1)
(a) Show that c=6\displaystyle c = 6
(4)
Given that (x+2)\displaystyle (x + 2) is a factor of f(x)\displaystyle f(x)
(b) show that the equation f(x)=0\displaystyle f(x) = 0 has only one real root.
(4)