Hint
Retain signs inside every bracket. Expand and collect \(\displaystyle x^{2}\) terms.
Worked solution
(a)
\(x^{2} - 14 x + 24\) multiplied by \(x + 4\) gives \(x^{3} - 10 x^{2} - 32 x + 96\).
Answer: \(x^{3} - 10 x^{2} - 32 x + 96\)
(b)
Collect all degree-one products.
\(x^{3} - 10 x^{2} - 32 x + 96\)
Answer: \(-32\)
(c)
Multiply the three constant terms.
\(96\)
Answer: \(96\)
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: Multiply the factors and collect each power of x.
- 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 multiply only matching powers.
(b) · 2 marks
- 1 mark: Collect all degree-one products.
- 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 polynomial identity is stronger than one equation at one x.
(c) · 2 marks
- 1 mark: Multiply the three constant terms.
- 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 multiply only matching powers.
Total: 6 marks.