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) $x = -4$, $\frac{3}{2}$, or $5$; (c) $y = \frac{1}{2}\lg\frac{3}{2}$ or $\frac{1}{2}\lg 5$

2023 ACSBR AM S3 EOY Q12

2023 ACSBR AM S3 EOY Q12

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