Hint
Use the leading coefficient when the polynomial is not monic. Read minus the next-to-leading coefficient of the monic polynomial.
Worked solution
(a)
\(\sum\alpha=27\)
Answer: \(27\).
Answer: \(27\)
(b)
\(\sum\alpha^2=(\sum\alpha)^2-2\sum_{i<j}\alpha_i\alpha_j=245\)
Answer: \(245\).
Answer: \(245\)
(c)
\(\displaystyle \sum \frac{1}{\alpha}= \frac{e_{n- 1}}{e_{n}}= \frac{121}{360}\)
Answer: \(\displaystyle \frac{121}{360}\).
Answer: \(\displaystyle \frac{121}{360}\)
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: Read minus the next-to-leading coefficient of the monic polynomial.
- 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: Keep the alternating coefficient signs.
(b) · 2 marks
- 1 mark: Subtract twice the pairwise-product sum from the squared root sum.
- 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 sum of squares includes cross terms when expanded.
(c) · 2 marks
- 1 mark: Divide the penultimate elementary symmetric sum by the 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: Keep the alternating coefficient signs.
Total: 6 marks.