1.2: Factors of a polynomial

4PM1/2/June/2021 — Question 6 · 13 marks

f(x)=x3+(p+1)x2−10x+q,f(x)=x^3+(p+1)x^2-10x+q,
where p\displaystyle p and q\displaystyle q are integers.
Given that (x−3)\displaystyle (x-3) is a factor of f(x)\displaystyle f(x),
(a) show that
9p+q+6=0.9p+q+6=0.
(3)
Given that (x+p)\displaystyle (x+p), where p>0\displaystyle p>0, is also a factor of f(x)\displaystyle f(x),
(b) show that
p2+10p+q=0.p^2+10p+q=0.
(3)
(c) Hence find the value of p\displaystyle p and the value of q\displaystyle q.
(5)
(d) Using your values of p\displaystyle p and q\displaystyle q, factorise f(x)\displaystyle f(x) completely.
(2)