Hint
The numerator degree is below the denominator degree. Clear denominators and substitute a linear-factor zero.
Worked solution
(a)
\(\displaystyle \frac{13 x^{2} + 97 x + 186}{x^{3} + 10 x^{2} + 32 x + 32} = \frac{2}{x + 4} - \frac{3}{\left(x + 4\right)^{2}} + \frac{11}{x + 2}\) follows after clearing the denominator and solving the three coefficient equations.
Answer: \(\displaystyle \frac{2}{x + 4} - \frac{3}{\left(x + 4\right)^{2}} + \frac{11}{x + 2}\)
(b)
Multiply each numerator by its missing denominator factors, add, and simplify to the original numerator.
Answer: \(\displaystyle \frac{13 x^{2} + 97 x + 186}{x^{3} + 10 x^{2} + 32 x + 32}\)
(c)
Use the remaining coefficient equation.
\(\displaystyle \frac{2}{x + 4} - \frac{3}{\left(x + 4\right)^{2}} + \frac{11}{x + 2}\)
Answer: \(-3\)
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.