Hint
The divisor has a nonzero leading coefficient. Divide the leading terms.
Worked solution
(a)
\(x^{3} + 6 x^{2} - 36 x - 5 = \left(x - 4\right) \left(x^{2} + 10 x + 4\right) + 11\) verifies the complete division identity.
Answer: \(x^{2} + 10 x + 4\); remainder \(11\)
(b)
At \(\displaystyle x=d\) the divisor term is zero, leaving exactly the constant remainder.
Answer: \(11\)
(c)
Continue until the remainder has degree zero.
\(x^{2} + 10 x + 4\)
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: Perform polynomial division and retain the complete lower-degree 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: Subtract the whole product, including its negative terms.
(b) · 2 marks
- 1 mark: Substitute d and relate P(d) to the remainder for division by x−d.
- 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: The remainder is not another quotient term.
(c) · 2 marks
- 1 mark: Continue until the remainder has degree zero.
- 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: Subtract the whole product, including its negative terms.
Total: 6 marks.