Question

Find the quadratic polynomial, the sum and product of whose zeroes are -5 and 6 respectively.

Answer 1

Find the quadratic polynomial:

As, $\alpha +ꞵ=-5\text{……………(i)}$ (given)
and $\alpha ꞵ=6\text{……………(ii)}$ (given)
Also, for the standard quadratic polynomial
$p\left(x\right)=a{x}^2+bx+c$, we have
$\alpha +ꞵ=-\frac{b}{a}\text{……………(iii)}$
and $\alpha ꞵ=\frac{c}{a}\text{……………(iv)}$
From (i), (ii), (iii), (iv), we have
$\alpha +ꞵ=-5$
$\alpha +ꞵ=-\frac{b}{a}$
$\Rightarrow \frac{b}{a}=5\text{……………(v)}$

and $\alpha ꞵ=6$

$\alpha ꞵ=\frac{c}{a}$
$\Rightarrow \frac{c}{a}=6.....\text{............(vi)}$
From (v) and (vi) ,we have
$a=1,b=5,c=6$

Now with the help of standard form of a polynomial
$p\left(x\right)=a{x}^2+bx+c$, we have
$p\left(x\right)=x^2+5x+6$,

Thus, the polynomial is $p\left(x\right)=x^2+5x+6$