Question

It being given that 2 is one of the zeros of the polynomial x³ − 4x² + x + 6. Find its other zeros.

Answer 1

$\begin{array}{l}\text{Let}f\left(x\right)=x^3-4x^2+x+6\\ \text{Since 2 is a zero of}f\left(x\right),(x-2)\text{is a factor of}f\left(x\right).\\ \text{On dividing}f\left(x\right)\text{by}(x-2),\text{we get:}\end{array}$



$\begin{array}{l}\\ f\left(x\right)=(x^3-4x^2+x+6)\\ =(x-2)(x^2-2x-3)\\ =(x-2)(x^2-3x+x-3)\\ =(x-2)\left\{x\right(x-3)+1(x-3\left)\right\}\\ =(x-2)(x-3)(x+1)\\ \therefore f\left(x\right)=0=>(x-2)=0\text{or}(x-3)=0\text{or}(x+1)=0\\ =>x=2\text{or}x=3\text{or}x=-1\\ \text{Thus, the other two zeroes are 3 and}-\text{1}\text{.}\end{array}$