Hint
The numerator degree is below the denominator degree. Clear denominators and substitute a linear-factor zero.
Worked solution
(a)
\(\displaystyle \frac{15 x^{2} + 5 x + 288}{x^{3} + 3 x^{2} + 25 x + 75} = \frac{3 x - 4}{x^{2} + 25} + \frac{12}{x + 3}\) follows after clearing the denominator and solving the three coefficient equations.
Answer: \(\displaystyle \frac{3 x - 4}{x^{2} + 25} + \frac{12}{x + 3}\)
(b)
Multiply each numerator by its missing denominator factors, add, and simplify to the original numerator.
Answer: \(\displaystyle \frac{15 x^{2} + 5 x + 288}{x^{3} + 3 x^{2} + 25 x + 75}\)
(c)
Use the remaining coefficient equation.
\(\displaystyle \frac{3 x - 4}{x^{2} + 25} + \frac{12}{x + 3}\)
Answer: \(-4\)
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: Use a complete decomposition form and determine all coefficients.
- 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 repeated factor needs both powers in the decomposition.
(b) · 2 marks
- 1 mark: Recombine the fractions over the original denominator and recover the numerator.
- 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 use only a constant above an irreducible quadratic.
(c) · 2 marks
- 1 mark: Use the remaining coefficient equation.
- 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 repeated factor needs both powers in the decomposition.
Total: 6 marks.