Conditions for a Quartic Equation to Have Three Equal Roots

In this article, we will explore the conditions under which a quartic equation of the form x^4 px^3 qx^2 rx s 0 might have three equal roots. We will investigate the algebraic steps and derivations that lead us to this conclusion.

Introduction to Quartic Equations

A quartic equation, also known as a polynomial of degree 4, takes the form:

[ x^4 px^3 qx^2 rx s 0 ]

For this equation to have three equal roots, it must have a repeated root of multiplicity 3. Let's denote one of these equal roots as ( r ). The equation can then be expressed as:

[ (x - r)^3 (x - k) 0 ]

where ( k ) is the fourth root, which may be equal to ( r ) or be a distinct root.

Expanding the Polynomial

We start by expanding the expression( (x - r)^3 (x - k) ).

Step 1: Expand ( (x - r)^3 ): ( (x - r)^3 x^3 - 3rx^2 3r^2x - r^3 ) Step 2: Expand ( (x - r)^3 (x - k) ): Multiplying ( x^3 - 3rx^2 3r^2x - r^3 ) by ( x - k ) ( (x^3 - 3rx^2 3r^2x - r^3) (x - k) x^4 - kx^3 - 3rx^3 3rkx^2 - 3r^2x^2 3r^2kx - r^3x kr^3 ) Step 3: Combine Like Terms: Combining the terms, we get: [ x^4 - (k 3r)x^3 (3rk 3r^2)x^2 - (3r^2k r^3)x kr^3 0 ]

Coefficients Comparison

To ensure that our polynomial matches the original form ( x^4 px^3 qx^2 rx s 0 ), we equate the coefficients:

( p -(k 3r) ) ( q 3rk 3r^2 ) ( r -(3r^2k r^3) ) ( s kr^3 )

Conditions for Three Equal Roots

For the polynomial to have three equal roots, the following conditions must be satisfied:

The first derivative must also have ( r ) as a root of at least multiplicity 1: Note: The first derivative of the polynomial is:

[ frac{d}{dx} (x^4 px^3 qx^2 rx s) 4x^3 3px^2 2qx r ]

Setting ( frac{d}{dx} (4x^3 3px^2 2qx r) 0 ), we get: [ 12x^2 6px 2q 0 ] The discriminant of this quadratic equation must be zero or a perfect square to ensure the presence of a repeated root at ( x r ). The second derivative must have ( r ) as a root of at least multiplicity 1: Note: The second derivative of the polynomial is:

[ frac{d^2}{dx^2} (x^4 px^3 qx^2 rx s) 12x^2 6px 2q ]

Setting ( frac{d^2}{dx^2} (12x^2 6px 2q) 0 ), we get: [ 24x 6p 0 ] Again, the discriminant of this linear equation must be zero or a perfect square to ensure the presence of a repeated root at ( x r ). Discriminant Condition: Note: The discriminant condition must be used to ensure that the polynomial has multiple roots. This is done by setting the discriminant of the original quartic equation to zero or a perfect square. The discriminant of the polynomial must be zero or a perfect square to ensure multiple roots.

Conclusion

To summarize, for the polynomial ( x^4 px^3 qx^2 rx s 0 ) to have three equal roots, it must satisfy the conditions outlined above regarding the coefficients and derivatives, ensuring the presence of a root of multiplicity at least 3. This rigorous approach guarantees that the polynomial meets the necessary criteria for having three equal roots.