Question

Find the zeros of the following polynomial by factorization method and verify the relations between the zeroes and the coefficient of the polynomials :
${4x}^2-3x-1$

Answer 1

Let $f\left(x\right)=4x^2-3x-1$

Splitting the middle term, we get

$=4x^2-4x+1x-1$

$=4x(x-1)+1(x-1)$

$=(x-1)(4x+1)$

For $f\left(x\right)=0$, we have

$4x^2-3x-1=0$

or $(x-1)(4x+1)=0$

Either $x-1=0\Rightarrow x=1$

or $4x+1=0$

$\Rightarrow 4x=-1$

$\Rightarrow x=\frac{-1}{4}$

$\therefore$ The zeroes of $f\left(x\right)$ are $1$ and $\frac{-1}{4}$.

Verification:

$\alpha =1,\beta =\frac{-1}{4}$

$a=4,b=-3$ and $c=-1$

$\therefore \alpha +\beta =\frac{-b}{a}$

$\Rightarrow 1-\frac{1}{4}=\frac{-(-3)}{4}$

$\Rightarrow \frac{3}{4}=\frac{3}{4}$

$\Rightarrow LHS=RHS$

Hence, verified

$\alpha .\beta =\frac{c}{a}$

$\Rightarrow 1\times \left(\frac{-1}{4}\right)=\frac{-1}{4}$

$\Rightarrow \frac{-1}{4}=\frac{-1}{4}$

$\Rightarrow LHS=RHS$

Hence, verified