1.1: Determine a polynomial and its real roots

4PM1/2R/June/2022 — Question 4 · 8 marks

f(x)=x3+px2+qx+7,f(x)=x^3+px^2+qx+7,
where p\displaystyle p and q\displaystyle q are integers. (x+1)\displaystyle (x+1) is a factor of f(x)\displaystyle f(x). The remainder when f(x)\displaystyle f(x) is divided by (x+2)\displaystyle (x+2) is −5\displaystyle -5.
(a) Find the value of p\displaystyle p and the value of q\displaystyle q.
(5)
(b) Hence show that f(x)=0\displaystyle f(x)=0 has only one real root.
(3)