Question

Find the zeroes of the following polynomials by factorisation method and verify the relation between the zeroes and the coefficients of the polynomials:

$3x^2+4x-4$.

Answer 1

Let $f\left(x\right)=3x^2+4x-4$.
Comparing it with the standard quadratic polynomial $a{x}^2+bx+c$, we get,
$a=3$, $b=4$, $c=-4$.
Now, $3x^2+4x-4$
$=3x^2+6x-2x-4$
$=3x(x+2)-2(x+2)$
$=(x+2)(3x-2)$.
The zeros of $f\left(x\right)$ are given by $f\left(x\right)=0$.
$=>(x+2)(3x-2)=0$
$=>x+2=0,3x-2=0$
$=>x=-2,x=\frac{2}{3}$.
Hence the zeros of the given quadratic polynomial are $-2$, $\frac{2}{3}$.

Verification of the relationship between the roots and the coefficients:
Sum of the roots $=-2+\frac{2}{3}$
$=\frac{-6+2}{3}$
$=\frac{-4}{3}$
$=\frac{-coefficientofx}{coefficientof{x}^2}$.
Product of the roots $=-2\times \left(\frac{2}{3}\right)$
$=\frac{-4}{3}$
$=\frac{constantterm}{coefficientof{x}^2}$.

Therefore, hence verified.