Question

It is given that −1 is one of the zeros of the polynomial x³ + 2x² − 11x − 12. Find all the given zeros of the given polynomial.

Answer 1

$\begin{array}{l}\text{Let}f\left(x\right)=x^3+2x^2-11x-12\\ \text{Since -1 is a zero of}f\left(x\right),(x+1)\text{is a factor of}f\left(x\right)\text{.}\\ \text{On dividing}f\left(x\right)\text{by}(x+1)\text{, we get:}\\ \end{array}$



$\begin{array}{l}\\ f\left(x\right)=x^3+2x^2-11x-12\\ =(x+1)(x^2+x-12)\\ =(x+1)\{x^2+4x-3x-12\}\\ =(x+1)\left\{x\right(x+4)-3(x+4\left)\right\}\\ =(x+1)(x-3)(x+4)\\ \therefore f\left(x\right)=0=>(x+1)(x-3)(x+4)=0\\ =>(x+1)=0\text{or}(x-3)=0\text{or}(x+4)=0\\ =>x=-1\mathrm{or}x=3\text{or}x=-4\\ \text{Thus, all the zeroes are}-1,3\mathrm{and}-4.\end{array}$