Hint
The polynomial coefficients are real. Use the conjugate quadratic factor and then the remaining linear factor.
Worked solution
(a)
Real coefficients give the conjugate a−bi. Their quadratic factor is \(\displaystyle (z-a)^{2}+b^{2}\); division leaves z−c.
Answer: \(\left\{4, 9 - 2 i, 9 + 2 i\right\}\)
(b)
Use the conjugate quadratic factor and then the remaining linear factor.
\(2\)
Answer: \(2\)
(c)
Use the conjugate quadratic factor and then the remaining linear factor.
\(85\)
Answer: \(85\)
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: Use conjugate pairing for the real-coefficient polynomial and find its remaining real factor.
- 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: Changing the sign of the whole root is not conjugation.
(b) · 2 marks
- 1 mark: Use the conjugate quadratic factor and then the remaining linear factor.
- 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: A negative discriminant does not mean there are no complex roots.
(c) · 2 marks
- 1 mark: Use the conjugate quadratic factor and then the remaining linear factor.
- 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: Changing the sign of the whole root is not conjugation.
Total: 6 marks.