Question

If 1 and −2 are two zeros of the polynomial (x³ − 4x² − 7x + 10), find its third zero.

Answer 1

$\begin{array}{l}\text{Let}f\left(x\right)=x^3-4x^2-7x+10\\ \text{Since 1 and}-2\text{are the zeroes of}f\left(x\right),\text{it follows that each one of}(x-1)\text{and}\\ (x+2)\text{is a factor of}f\left(x\right).\\ \text{Consequently,}(x-1)(x+2)=(x^2+x-2)\text{is a factor of}f\left(x\right).\\ \text{On dividing}f\left(x\right)\text{by}(x^2+x-2)\text{, we get:}\end{array}$



$\begin{array}{l}f\left(x\right)=0=>(x^2+x-2)(x-5)=0\\ =>(x-1)(x+2)(x-5)=0\\ =>x=1\text{or}x=-2\text{or}x=5\\ \text{Hence, the third zero is 5}\text{.}\end{array}$