Question

Find the zeros of the following polynomial by factorization method and verify the relations between the zeroes and the coefficient of the polynomials :
${5t}^2+12t+7$

Answer 1

Let $f\left(t\right)=5t^2+12t+7$

For zeroes of $f\left(t\right),f\left(t\right)=0$

$\Rightarrow 5t^2+12t+7=0$

$\Rightarrow 5t^2+7t+5t+7=0$

$\Rightarrow t(5t+7)+1(5t+7)=0$

$\Rightarrow (5t+7)(t+1)=0$

$\Rightarrow 5t+7=0$ or $(t+1)=0$

$\Rightarrow t=\frac{-7}{5}$ or $t=-1$

verification : $\alpha =-\frac{7}{5}\beta =-1$

$a=5,b=12,c=+7$

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

$\Rightarrow \frac{7}{5}-1=\frac{-(+12)}{5}$

$\Rightarrow \frac{-7-5}{5}=\frac{-12}{5}$

$\Rightarrow \frac{-12}{5}=\frac{-12}{5}$

$LHS=RHS$

Hence, verified.

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

$\Rightarrow \left(\frac{-7}{5}\right)(-1)=\frac{7}{5}$

$\Rightarrow +\frac{7}{5}=\frac{7}{5}$

$\Rightarrow LHS=RHS$

Hence, verified.