2024 ACS(BR) P2 Q6

2024 ACS(BR) P2 Q6

Secondary 4
9 marks
    1. Using the substitution \(u=x^3\) or otherwise, express \(x^6-1\) as the product of two factors.[1]
    2. Hence express \(x^6-1\) as the product of four factors with integer coefficients.[1]
    1. Find the remainder when \(\mathrm{f}(x)=3x^3-5x^2+7x-4\) is divided by \(x-1\).[1]
    2. Hence show that \(h=-1\) for which \(\mathrm{g}(x)=\mathrm{f}(x)+h\) is divisible by \(x-1\).[2]
    3. Explain why the equation \(\mathrm{g}(x)=0\) has only one real root.[4]

Solution:

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Answer:(a)(i) \((x^3-1)(x^3+1)\). (a)(ii) \((x-1)(x+1)(x^2+x+1)(x^2-x+1)\). (b)(i) \(1\). (b)(ii) \(h=-1\). (b)(iii) Only \(x=1\); the quadratic factor has discriminant \(-56<0\).

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