2025 Presbyterian High School Prelims A Math Paper 2
Factorise \(x^3 - 125\).[1]
The expression \(-2x^3 + 7x^2 + ax + b\), where \(a\) and \(b\) are integers, has a factor of \(x - 5\) and a remainder of \(-11\) when divided by \(2x + 1\). Find the value of \(a\) and of \(b\).[4]
Hence, or otherwise, show that the equation \(x^3 - 125 = -2x^3 + 7x^2 + ax + b\) has only one real root.[4]
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Answer:(a) \((x-5)(x^2+5x+25)\) (b) \(a=16,\ b=-5\) (c) Only one real root, \(x=5\)