Hint

Pair four of the (1+x) factors with the four (1−x) factors. Evaluations at ±1 separate even and odd coefficients.

Worked solution

(a)

Split (1+x)⁶ into (1+x)²(1+x)⁴. Then combine the two fourth powers using (1+x)(1−x)=1−x². This is a polynomial identity, so no division or excluded x is introduced.

Answer: \(G(x)=(1+x)^2\{(1+x)(1-x)\}^4=(1+x)^2(1-x^2)^4\).

(b)

The even core begins \((1-x^2)^4=1-4x^2+6x^4-4x^6+x^8\). Multiplying by 1+2x+x², an x⁵ contribution can only come from (2x)(6x⁴), so the coefficient is 12.

Answer: 12.

(c)

Let E and O be the even and odd coefficient sums. G(1)=E+O=0 and G(−1)=E−O=0, because the original product has a zero factor at each input. Thus \(O=(G(1)-G(-1))/2=0\).

Answer: 0.

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) · 2 marks

• 1 mark: Separate two factors of (1+x) from its sixth power.
• 1 mark: Use the difference-of-squares identity for the remaining fourth powers.

Alternative methods: Multiplying both sides to the same polynomial is accepted.

Common error: Replacing the entire product by (1−x²)⁶ and losing the exponent imbalance.

(b) · 3 marks

• 1 mark: Obtain the core x⁴ coefficient 6.
• 1 mark: Identify 2x times 6x⁴ as the only contributing pair.
• 1 mark: Give 12.

Alternative methods: Convolution from the original two factors is accepted if all contributions are included.

Common error: Assuming the whole product has only even powers because its core does.

(c) · 3 marks

• 1 mark: Relate the odd coefficient sum to (G(1)−G(−1))/2.
• 1 mark: Evaluate both endpoint values as zero.
• 1 mark: Conclude that the odd coefficient sum is zero.

Alternative methods: Adding all odd coefficients of a full correct expansion is accepted.

Common error: Using G(0), which gives only the constant coefficient.

Total: 8 marks.