When the polynomial ${{x}^{3}}+a{{x}^{2}}+bx+c$ is divided by $\left( x-1 \right)$, \(\left( x-2

Question

When the polynomial ${{x}^{3}}+a{{x}^{2}}+bx+c$ is divided by $\left( x-1 \right)$, $\left( x-2 \right)$ and $\left( x-3 \right)$, the remainders are $8$, $12$ and $25$ respectively. Find the values of $a$, $b$ and $c$.[4] A curve has equation $y=\mathrm{f}\left( x \right)$, where $\mathrm{f}\left( x \right)={{x}^{3}}+a{{x}^{2}}+bx+c$, with the values of $a$, $b$ and $c$ found in part (i). Show that the gradient of the curve is always positive. Hence explain why the equation $\mathrm{f}\left( x \right)=0$ has only one real root and find this root.[3] Find the $x$- coordinates of the points where the tangent to the curve is parallel to the line $y=2x-3$.[3]

Accepted Answer

(i) $a = -\frac{3}{2}$, $b = \frac{3}{2}$, $c = 7$ (iii) $1.15$ ($3$ s.f.) and $-0.145$ ($3$ s.f.)

N2017 P1 Q5

N2017 P1 Q5

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