Question

. If alpha and beta are zeroes of a quadratic polynomial p(x), then factorize p(x

Answer 1

let p(x)=ax²+bx+c
we can use spit the middle term
so
α+β=-b/a
and
αβ=c/a
x=αorβ
for more give the equation please

Answer 2

Polynomials

Solution.

If $\alpha$ and $\beta$ are the zeroes of a quadratic polynomial, then the quadratic polynomial can be written as

$\quad p(x)=x^{2}-(\alpha+\beta)x+\alpha\beta$

[ We have used the relation between zeroes and coefficients above. ]

We have:

$\quad p(x)=x^{2}-(\alpha+\beta)x+\alpha\beta$

$\Rightarrow p(x)=x^{2}-\alpha x-\beta x+\alpha\beta$

$\Rightarrow p(x)=x(x-\alpha)-\beta(x-\alpha)$

$\Rightarrow p(x)=(x-\alpha)(x-\beta)$

This is the required factorization.

Remark. The problem could have a direct answer $p(x)=(x-\alpha)(x-\beta)$ where $\alpha$ and $\beta$ are the zeroes of $p(x)$, but that does not serve our purpose here.