Hint
The scale a is nonzero. Substitute \(\displaystyle x= \frac{y- b}{a}\) and clear \(\displaystyle a^{3}\).
Worked solution
(a)
\(y^{3} - 108 y^{2} + 3867 y - 45920=0\)
Answer: \(-108\).
Answer: \(-108\)
(b)
\(B=3867\)
Answer: \(3867\).
Answer: \(3867\)
(c)
\(C=-\prod(a\alpha+b)=-45920\)
Answer: \(-45920\).
Answer: \(-45920\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Substitute \(\displaystyle x= \frac{y- b}{a}\) and clear \(\displaystyle a^{3}\).
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not substitute \(\displaystyle y=ax+b\) in the wrong direction.
(b) · 2 marks
- 1 mark: Expand the transformed monic polynomial and collect the y term.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A shift b is added once per root when finding the new root sum.
(c) · 2 marks
- 1 mark: Read the signed transformed root product.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not substitute \(\displaystyle y=ax+b\) in the wrong direction.
Total: 6 marks.