Higher Maths Unit Assessment Revision
Relationships and Calculus
1.1 Polynomials
1) Show that (x + 1) is a factor of 𝑓(𝑥) = 𝑥 3 + 4𝑥 2 − 7𝑥 − 10 and hence fully factorise
𝑓(𝑥).
2) Show that (x – 3) is a factor of 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 10𝑥 − 24 and hence fully factorise
𝑓(𝑥).
3) Show that (x – 4) is a factor of 𝑓(𝑥) = 3𝑥 3 − 𝑥 2 − 38𝑥 − 24 and hence fully factorise
𝑓(𝑥).
4) Show that (x + 3) is a factor of 𝑓(𝑥) = 6𝑥 3 + 13𝑥 2 − 19𝑥 − 12 and hence fully factorise
𝑓(𝑥).
5) Show that (2x – 5) is a factor of 𝑓(𝑥) = 12𝑥 3 − 63𝑥 − 30 and hence fully factorise 𝑓(𝑥).
6) If the function 𝑔(𝑥) = 𝑘𝑥 2 + 3𝑥 − 1 has no real roots then calculate the range of values
for k.
7) If the function ℎ(𝑥) = 2𝑥 2 + 𝑘𝑥 + 7 has 1 real root then calculate the range of values for
k.
8) If the function 𝑓(𝑥) = 3𝑥 2 − 4𝑥 − 3 + 𝑘 has 2 distinct real roots then calculate the range
of values for k.
9) If (x + 5) is a factor of the function 𝑓(𝑥) = 𝑥 3 + 8𝑥 2 + 𝑝𝑥 − 140 then calculate the range
of values for p.
10) If (x – 7) is a factor of the function 𝑔(𝑥) = 2𝑥 3 + 𝑞𝑥 2 − 10𝑥 + 21 then calculate the
range of values for q.