Question

if there is no'x' term in a cubic polynomial then alpha*beta*gamma =0 is it true?​

Answer 1

Cubic polynomial

Let us know a cubic polynomial $f(x)$ such that

$\quad f(x)=ax^{3}+bx^{2}+cx+d$

Consider the zeroes $\alpha,\beta,\gamma$. Then

• $\alpha+\beta+\gamma=-\frac{b}{a}$
• $\alpha\beta+\beta\gamma+\gamma\alpha=\frac{c}{a}$
• $\alpha\beta\gamma=-\frac{d}{a}$

Let us move to the solution now.

Now when the $x-term$ does not exist, we must take its coefficient being $0\:(zero).$

From $f(x)$, we can write $c=0.$

Here the product of the zeroes $\alpha,\beta,\gamma$ i.e. $\alpha\beta\gamma$ does not depend on $c$ and thus we can not give nod to the certainty of the product being $0\:(zero).$

The only possible case for $\alpha\beta\gamma=0$ is when $a\neq 0,d=0.$

Answer 2

Zeroes of a polynomial means its roots

So let α,β,γ be the roots of a cubic polynomial

We can express the polynomial in two forms

1.From the roots we can write a polynomial as

( x- α)(x- β)(x- γ )=0

2.From knowledge of relation between coefficient and roots .

Let the polynomial be ax³+bx²+cx+d = 0

We know

Σα = α+β+γ = -b/a ;

Σαβ= αβ+βγ+αγ=c/a;

Σαβγ =αβγ =-d/a.

So the above polynomial is written as

x³+b/ax²+c/ax+d/a = 0

x³- (α+β+γ)x² +(αβ+βγ+αγ)x - (αβγ) =0

That's it. Even the first one on expansion will give the same result as second one .

This is the way to represent a polynomial,whose roots are known.

I hope it helps uh akka ✨