Condition for this polynomial to be a perfect square

In summary, in order for a fourth degree polynomial to be a perfect square, it must be the product of two identical second degree polynomials with roots α and β. This means that two of the roots of the polynomial must be identical. The general form of a second degree polynomial is (x^2+mx+n)^2, which can be expanded and compared to the given fourth degree polynomial to find the conditions on a, b, c, and d for it to be a perfect square.
  • #1
utkarshakash
Gold Member
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Homework Statement


The condition that [itex]x^4+ax^3+bx^2+cx+d [/itex] 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

[itex](x-\alpha)^2(x-\beta)^2[/itex] 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?
 
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  • #2
utkarshakash said:

Homework Statement


The condition that [itex]x^4+ax^3+bx^2+cx+d [/itex] 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

[itex](x-\alpha)^2(x-\beta)^2[/itex] 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

[tex](x^2+mx+n)^2[/tex]

And once you expand this out, you can compare coefficients and find the conditions on a,b,c,d as required.
 
  • #3
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.
 

1. What is a perfect square polynomial?

A perfect square polynomial is a polynomial that can be written as the square of another polynomial. In other words, it is the product of two identical factors.

2. What is the condition for a polynomial to be a perfect square?

The condition for a polynomial to be a perfect square is that all of its terms must have even exponents. This means that when the polynomial is expanded, all of the terms will have a coefficient that is a perfect square.

3. How do you check if a polynomial is a perfect square?

To check if a polynomial is a perfect square, you can expand it and see if all of the terms have even exponents. Alternatively, you can also take the square root of the polynomial and see if it can be simplified to a polynomial with integer coefficients.

4. Can a polynomial with odd exponents be a perfect square?

No, a polynomial with odd exponents cannot be a perfect square. This is because when expanded, at least one term will have an odd exponent and therefore will not be a perfect square.

5. How do perfect square polynomials relate to quadratic equations?

A perfect square polynomial can be used to solve quadratic equations. This is because when a quadratic equation is written in the form of a perfect square polynomial, it can easily be factored into two identical factors, making it easier to find the solutions.

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