The function $\mathrm{f}(x) = 2x^3 + ax^2 + bx + 60$, where $a$ and $b$ are constants, is exac

Question

The function $\mathrm{f}(x) = 2x^3 + ax^2 + bx + 60$, where $a$ and $b$ are constants, is exactly divisible by $x + 4$. The remainder is $-18$ when $\mathrm{f}(x)$ is divided by $x - 2$. Find the value of $a$ and of $b$.[4] By showing your working clearly, solve $\mathrm{f}(x) = 0$.[4] Hence solve the equation $2(10^{6y}) + a(10^{4y}) + b(10^{2y}) = -60$, where $a$ and $b$ are the values found in part (a).[3]

Accepted Answer

(a) $a=-5$, $b=-37$ (b) $x=-4,\ \dfrac32,\ 5$ (c) $y=\dfrac12\lg\left(\dfrac32\right)$ or $\dfrac12\lg5$

2023 ACSBR AM S3 EOY Q12

2023 ACSBR AM S3 EOY Q12

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