Hint
Use the zero of the factor, with its correct sign. Use the given factor to set \(\displaystyle P(a)=0\).
Worked solution
(a)
\(3 k + 603 = 0\) determines k uniquely since \(\displaystyle a\ne 0\).
Answer: \(-201\)
(b)
Divide out x−a, factor the quadratic and set every factor equal to zero.
Answer: \(\left(x - 13\right) \left(x - 3\right) \left(x + 15\right)\); \(\left\{-15, 3, 13\right\}\)
(c)
Keep the sign in the factor x+c.
\(-15\)
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: Substitute the stated factor root into P and solve for k.
- 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: A factor x+a corresponds to input −a.
(b) · 2 marks
- 1 mark: Divide out the known factor and factor the remaining polynomial.
- 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: One known factor does not complete a cubic solution.
(c) · 2 marks
- 1 mark: Keep the sign in the factor x+c.
- 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 factor x+a corresponds to input −a.
Total: 6 marks.