Hint
The odd-power signs change in B. For the product, combine corresponding factors before expanding.
Worked solution
(a)
The Pascal row is 1,5,10,10,5,1. All terms of A have positive coefficients. In B, the odd powers acquire minus signs and the even powers retain plus signs, giving the displayed expansions.
Answer: \(A=1+5x+10x^2+10x^3+5x^4+x^5\); \(B=1-5x+10x^2-10x^3+5x^4-x^5\).
(b)
Subtracting cancels the even powers and doubles the odd powers. Division by 2 gives \(5x+10x^3+x^5\).
Answer: \(5x+10x^3+x^5\).
(c)
\(A(x)B(x)=\{(1+x)(1-x)\}^5=(1-x^2)^5\). Its \(\displaystyle x^{2}\) term is \(5(-x^2)=-5x^2\), so the coefficient is −5. Directly, \(\displaystyle 10-25+10=-5\), including the cross-product.
Answer: −5.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Use the complete correct Pascal row through degree 5.
- 1 mark: Give the correct complete expansion of A.
- 1 mark: Give the correct complete expansion of B, including every odd-power sign.
Alternative methods: Repeated multiplication of brackets is accepted.
Common error: Stopping after three terms when the full expansion is requested.
(b) · 2 marks
- 1 mark: Cancel the even powers and retain the odd powers with doubled coefficients before division.
- 1 mark: Give \(\displaystyle 5x+10x^{3}+x^{5}\).
Alternative methods: Using the odd-part identity for a polynomial is accepted.
Common error: Cancelling the odd terms as if the two expressions were being added.
(c) · 3 marks
- 1 mark: Use the product identity \(\displaystyle (1-x^{2})^{5}\) or list all contributing power pairs.
- 1 mark: Obtain the \(\displaystyle x^{2}\) contribution with its correct sign.
- 1 mark: Give the coefficient −5.
Alternative methods: Multiplying only the necessary terms of A and B gives the same coefficient.
Common error: Adding the \(\displaystyle x^{2}\) coefficients 10+10 and missing the product of the linear terms.
Total: 8 marks.