Question

Question 1 (v)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials

(v) $2x^2+\frac{7}{2}x+\frac{3}{4}$

Answer 1

v) Let $f\left(x\right)=2x^2+\frac{7}{2}x+\frac{3}{4}=8x^2+14x+3$
$=8x^2+12x+2x+3$
$=4x(2x+3)+1(2x+3)$
$=(2x+3)(4x+1)$
So, the value of $8x^2+14x+3$ is zero when 2x + 3 = 0 or 4x + 1 = 0
i.e., when $x=-\frac{3}{2}orx=-\frac{1}{4}$
so, the zeroes of $8x^2+14x+3are-\frac{3}{2}and-\frac{1}{4}$
$\therefore$ sum of zeroes $=-\frac{3}{2}-\frac{1}{4}=-\frac{7}{4}=\frac{-7}{2\times 2}$
$=-\frac{\left(Coefficientofx\right)}{\left(Coefficientof{x}^2\right)}$
And product of zeroes $=(-\frac{3}{2})(-\frac{1}{4})=\frac{3}{8}=\frac{3}{2\times 4}$
$=\frac{Constantterm}{Coefficientof{x}^2}$
Hence, verified the relations between the zeroes and the coefficients of the polynomial.