Hint
Use \(\displaystyle \frac{b}{a}\) for divisor ax−b. Substitute \(\displaystyle x= \frac{1}{a}\).
Worked solution
(a)
\(\displaystyle x = \frac{1}{2}\) and \(\displaystyle x = - \frac{1}{3}\) are the zeros of the two divisors; evaluate P at those inputs.
Answer: \(\displaystyle \frac{123}{8}\) and \(\displaystyle \frac{287}{27}\)
(b)
\(\displaystyle x^{3} + 9 x^{2} + 4 x + 11 = \left(2 x - 1\right) \left(\frac{x^{2}}{2} + \frac{19 x}{4} + \frac{35}{8}\right) + \frac{123}{8}\) verifies the division.
Answer: \(\displaystyle \frac{x^{2}}{2} + \frac{19 x}{4} + \frac{35}{8}\); \(\displaystyle \frac{123}{8}\)
(c)
Divide \(\displaystyle x^{3}\) by ax.
\(\displaystyle \frac{1}{2}\)
Answer: \(\displaystyle \frac{1}{2}\)
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: Evaluate P at each divisor root, including the nonmonic scaling.
- 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 substitute b when the divisor is ax−b.
(b) · 2 marks
- 1 mark: Divide by ax−1 and retain the quotient and constant remainder.
- 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 divisor scale does not scale the remainder.
(c) · 2 marks
- 1 mark: Divide \(\displaystyle x^{3}\) by ax.
- 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 substitute b when the divisor is ax−b.
Total: 6 marks.