Hint
The three roots in these problems are distinct and simple. Order the roots and test interval signs.
Worked solution
(a)
The signs from left to right are −,+,−,+; include all three zero boundaries.
Answer: \(\left[8, 10\right] \cup \left[15, \infty\right)\)
(b)
Take precisely the negative intervals and exclude their zero boundaries.
Answer: \(\left(-\infty, 8\right) \cup \left(10, 15\right)\)
(c)
Set the last linear factor to zero.
\(15\)
Answer: \(15\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Find all three simple roots and use the cubic sign chart with included zeros.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not assume a quadratic sign pattern for a cubic.
(b) · 2 marks
- 1 mark: Use the alternating signs and exclude all equality endpoints.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Multiple roots do not necessarily reverse the sign.
(c) · 2 marks
- 1 mark: Set the last linear factor to zero.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not assume a quadratic sign pattern for a cubic.
Total: 6 marks.