Question

Find the zeroes of the quadratic polynomial $x^2+7x+12$ and verify the relationship between the zeroes and the coefficients of the polynomial.

Answer 1

Step 1: Form the equation

The given polynomial is $x^2+7x+12$

We know that the zeroes of a polynomial are evaluated by equating them with zero.

$\therefore p\left(x\right)=0$

$\Rightarrow x^2+7x+12=0$

Step 2: Solve to find the zeroes

$x^2+7x+12=0$

$\Rightarrow x^2+4x+3x+12=0$

$\Rightarrow$$x\left(x+4\right)+3\left(x+4\right)=0$

$\Rightarrow$ $\left(x+4\right)\left(x+3\right)=0$

$\Rightarrow$ $x=-4,-3$

Step 3: Verification

We know that for a given polynomial $a{x}^2+bx+c$,

Sum of the zeroes$=-\frac{b}{a}$ and product of the roots$=\frac{c}{a}$

Sum of the zeroes $=-4-3=-7$

Again, $-\frac{b}{a}=-\frac{7}{1}=-7$

Product of the zeroes $=\left(-4\right)\times \left(-3\right)=12$

Again, $\frac{c}{a}=\frac{12}{1}=12$

Thus, the relationship between the zeroes and the coefficients of the polynomial is verified.

Hence, the zeroes of this polynomial are $-4$ and $-3$.