Question

Find the zeros of the quadratic polynomial 5x² − 4 − 8x and verify the relationship between the zeros and the coefficients of the given polynomial.

Answer 1

$\begin{array}{l}\text{We have:}\\ f\left(x\right)=5x^2-4-8x\\ =5x^2-8x-4\\ \begin{array}{l}=5x^2-(10x-2x)-4\\ =5x^2-10x+2x-4\\ =5x(x-2)+2(x-2)\\ =(5x+2)(x-2)\\ \therefore f\left(x\right)=0=>(5x+2)(x-2)=0\\ =>5x+2=0\text{or}x-2=0\\ =>x=\frac{-2}{5}\text{or}x=2\\ \text{So, the zeros of}f\text{(}x\text{) are}\frac{-2}{5}\text{and}2.\end{array}\\ \text{Sum of the zeros}=\left(\frac{-2}{5}\right)+2=\frac{-2+10}{5}=\frac{8}{5}=\frac{-\left(\text{coefficient of}x\right)}{\left(\text{coefficient of}{x}^{\text{2}}\right)}\\ \text{Product of the zeros=}\left(\frac{-2}{5}\right)\times \text{2 =}\frac{-4}{5}=\frac{\text{constant term}}{\text{(coefficient of}{x}^{\text{2}})}\end{array}$