Question

Find quadratic polynimial, the sum and the product of whose zeros are $-\frac{8}{3}$ and $\frac{4}{3}$ respectively. Also find zeros of the polynomial by factorization.

Answer 1

$\Rightarrow$ Sum of the zeros $=\frac{-8}{3}$

$\Rightarrow$ Product of zeros $=\frac{4}{3}$

$\Rightarrow$ $x^2-\left(Sumofzeros\right)x+\left(Productofzeros\right)=0$

$\Rightarrow$ $x^2-\left(\frac{-8}{3}\right)x+\left(\frac{4}{3}\right)=0$

$\Rightarrow$ $x^2+\frac{8}{3}x+\frac{4}{3}=0$

$\Rightarrow$ $3x^2+8x+4=0$

$\therefore$ The required quadratic equation is $3x^2+8x+4=0$

Now, we are going to find zeros,

$\Rightarrow$ $3x^2+8x+4=0$

$\Rightarrow$ $3x^2+6x+2x+4=0$

$\Rightarrow$ $3x(x+2)+2(x+2)=0$

$\Rightarrow$ $(x+2)(3x+2)=0$

$\Rightarrow$ $x+2=0$ and $3x+2=0$

$\therefore$ $x=-2,\frac{-2}{3}$