Form factored of the polynomial discriminant

In summary, the polynomial discriminant for the equation x² - 2Ax + B² is A² - B², with a factored form of (A+B)(A-B). However, for the cubic equation x³ - 3Ax² + 3B²x - C³, the discriminant is more complex and does not have a simple factorization. The formula for the discriminant of a cubic can be used to find the value, but it may not be easily factorizable.
  • #1
Bruno Tolentino
97
0
I wrote x² - (a + b)x + (ab) in the wolfram and polynomial discriminant was: a² - 2ab + b². Factoring: (a-b)²

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So, I wrote x³ - (a+b+c) x² + (bc+ca+ab) x - (abc) and the polynomial discrimant given was:
31321321321321.png
Factoring: (b-c)² (c-a)² (a-b)²

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Now, I wrote x² - 2Ax + B² and the polynomial discriminant is: A² - B². Factoring: (A+B)(A-B)

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At last, but, no minus important, given this polynomial: x³ - 3 A x² + 3 B² x - C³, which is the factored form of the polynomial discriminant?
 
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  • #2
The formula for the discriminant of a cubic is here. Plug your coefficients into that and see what happens.

I see no reason to expect that an easy factorisation is available. The three preceding examples you gave are all simple. The first two polynomials are just ##(x-a)(x-b)## and ##(x-a)(x-b)(x-c)## respectively, and the third one is only a quadratic. The fourth, about which you ask, is something else entirely.
 
  • #3
andrewkirk said:
The formula for the discriminant of a cubic is here. Plug your coefficients into that and see what happens.
I know how find it! The problem is the factorization...
 

Related to Form factored of the polynomial discriminant

What is the form factor of a polynomial discriminant?

The form factor of a polynomial discriminant is a mathematical expression that represents the shape and characteristics of a polynomial function. It is typically written in the form of axn + bxn-1 + ... + k, where n is the degree of the polynomial and a, b, and k are constants.

Why is the form factor of a polynomial discriminant important?

The form factor of a polynomial discriminant is important because it helps us understand the behavior of a polynomial function. It can tell us about the number of roots the function has, whether those roots are real or complex, and the multiplicity of each root.

How do you find the form factor of a polynomial discriminant?

The form factor of a polynomial discriminant can be found by factoring the polynomial function. This can be done using various methods such as the quadratic formula, synthetic division, or grouping terms. Once the polynomial is factored, the form factor can be written in its standard form.

Can the form factor of a polynomial discriminant be negative?

Yes, the form factor of a polynomial discriminant can be negative. This typically occurs when the polynomial has an odd degree and has a negative coefficient for the highest degree term. In this case, the form factor will have a negative leading coefficient.

How is the form factor of a polynomial discriminant used in solving equations?

The form factor of a polynomial discriminant is used in solving equations by helping us determine the number and nature of the roots of a polynomial function. This information can then be used to find the solutions to the equations using methods such as factoring, the quadratic formula, or graphing.

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