Condition for this polynomial to be a perfect square

[utkarshakash]

Homework Statement

The condition that $x^4+ax^3+bx^2+cx+d$ is a perfect square, is

Homework Equations

The Attempt at a Solution

If the above polynomial will be a perfect square then it can be represented as

$(x-\alpha)^2(x-\beta)^2$ where α and β are the roots of it.This means that two roots of it will be identical.
Am I correct in my assumption?

 

[Mentallic]

Homework Statement

The condition that $x^4+ax^3+bx^2+cx+d$ is a perfect square, is

Homework Equations

The Attempt at a Solution

If the above polynomial will be a perfect square then it can be represented as

$(x-\alpha)^2(x-\beta)^2$ where α and β are the roots of it.This means that two roots of it will be identical.
Am I correct in my assumption?

Why do the 2 roots have to be identical? You've already shown what the general form of a quartic (degree 4 polynomial) needs to be in for it to be a perfect square, so go ahead and expand.

Although, I'd suggest you make the general form a general quadratic squared, so you'll have

$$(x^2+mx+n)^2$$

And once you expand this out, you can compare coefficients and find the conditions on a,b,c,d as required.

 

[aralbrec]

What does it mean to be a perfect square?

If the equation is quadratic ax^2+bx+c then it is a perfect square if it can be factored as (x+k)^2, ie two identical factors.

If you're starting with a fourth degree polynomial and you want to constrain it to be a perfect square then it must be the product of two identical factors. That factor must be a polynomial of degree 2 since you need an x^4 term when it is multiplied out. The most general form of a second degree polynomial is given above.