Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
$ 4 x^{2}-3 x-1 $

Given:

$4 x^{2}-3 x-1$

To find:

Here, we have to find the zeros of the given polynomial. 

Solution:

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

To find the zeros of f(x), we have to put $f(x)=0$.

This implies,

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

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

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

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

$4x+1=0$ and $x-1=0$

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

Therefore, the zeros of the quadratic equation $f(x) = 4x^2 - 3x - 1$ are $\frac{-1}{4}$ and $1$.

Verification:

We know that, 

Sum of zeros $= -\frac{\text { coefficient of x }}{\text { Coefficient of } \mathrm{x}^{2}}$

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

$=\frac{3}{4}$

Sum of the zeros of $f(x)=\frac{-1}{4}+1$

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

$=\frac{3}{4}$ 

Product of roots $= \frac{\text { constant term }}{\text { Coefficient of } \mathrm{x}^{2}}$

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

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

Product of the roots of $f(x)=\frac{-1}{4}\times1 =-\frac{1}{4}$

Hence, the relationship between the zeros and their coefficients is verified.

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Updated on: 10-Oct-2022

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