It is given that $\mathrm{f}(x) = x^3 + ax^2 - 5x + b$, where $a$ and $b$ are consta

Question

It is given that $\mathrm{f}(x) = x^3 + ax^2 - 5x + b$, where $a$ and $b$ are constants, has a factor of $x + 1$ and leaves a remainder of $-24$ when divided by $(x + 3)$. Show that $a = -1$ and $b = -3$.[4] Hence factorise $\mathrm{f}(x)$ completely and write down the solutions for $\mathrm{f}(x) = 0$.[3] From part (a), it is known that $\mathrm{f}(x) = x^3 - x^2 - 5x - 3$. The diagram below shows part of the curve $y = \mathrm{f}(x)$ and a line that cuts the curve at $B$ and $C$. $B$ has coordinates $(1, -6)$. $A$ and $C$ are points on the curve and the $x$-axis. Find the area of the shaded region.[5] Show that the equation of the tangent at the minimum point of the curve is $y = -\frac{256}{27}$.[3]

Accepted Answer

(a) $a = -1, b = -3$ (b) $\mathrm{f}(x) = (x + 1)^2(x - 3), x = -1$ or $x = 3$ (c) $12\frac{2}{3}$ units$^2$

2024 CGSS P2 Q7

2024 CGSS P2 Q7