Maths Question:
$\begin{align} & \text{Given that }x-1\text{ and }x-2\text{ are factors of the polynomial }{{x}^{3}}+a{{x}^{2}}+bx-6.\text{ } \\ & \text{Find }a\text{ and }b \\\end{align}$
Maths Solution:
$\begin{align} & \text{let }f(x)={{x}^{3}}+a{{x}^{2}}+bx-6 \\ & \text{Since (}x-1)\text{ and }(x-2)\text{ are factors, then }f(1)=0 \\ & f(2)=0 \\ & f(1)={{1}^{3}}+a{{(1)}^{2}}+b(1)-6=0 \\ & a+b=5----(i) \\ & f(2)={{2}^{3}}+a{{(2)}^{2}}+b(2)-6=0 \\ & 4a+2b=-2-----(ii) \\ & \text{Solving the (i) and (ii) simultaneously} \\ & \text{gives }a=-6,\text{ }b=11 \\\end{align}$
University mathstopic: