en.wikipedia.org/wiki/Quartic_equation
4 corrections found
proving that all attempts at solving the higher order polynomials would be futile.
The Abel–Ruffini theorem does not say that all higher-degree polynomials are unsolvable. It says only that there is no formula by radicals for the general polynomial of degree 5 or higher, while some specific higher-degree polynomials are solvable by radicals.
Full reasoning
This sentence overstates what the Abel–Ruffini theorem proves.
The theorem concerns the general polynomial of degree greater than 4: there is no single formula in radicals that solves every such equation from its coefficients. It does not imply that every polynomial of degree 5 or more is unsolvable by radicals.
A standard counterpoint is that many specific higher-degree equations are solvable by radicals. The Encyclopedia of Mathematics states that "the general equation of degree (n>4) is unsolvable by radicals," but immediately adds that this "does not contradict the fact that some algebraic equations ... are solvable by radicals" and that "Some special equations of degree (n) are solvable by radicals." Treccani likewise notes that Abel–Ruffini "does not assert that no polynomial of degree greater than 4 is solvable by radicals," giving polynomials of the form (x^n-1) as a counterexample.
So the quoted claim is incorrect because it turns a theorem about the general case into a statement about all higher-order polynomials.
2 sources
- Algebraic equation - Encyclopedia of Mathematics
In other words, the general equation of degree n > 4 is unsolvable by radicals. However, Abel's theorem does not contradict the fact that some algebraic equations with numerical coefficients ... are solvable by radicals. Some special equations of degree n are solvable by radicals.
- Abel-Ruffini, teorema di - Enciclopedia della Matematica (Treccani)
È bene sottolineare che il teorema di Abel-Ruffini non afferma che nessun polinomio di grado maggiore di 4 è risolubile per radicali ... Afferma invece che, per ogni grado n > 4, esistono polinomi di grado n che non sono risolubili per radicali.
where l and m are two different real numbers.
A quartic with a triple root and a simple root does not need both roots to be real. That restriction only follows for real-coefficient quartics, but the text states it as a general quartic case.
Full reasoning
This is not correct for a general quartic equation.
Polynomial coefficients may be complex as well as real. The Encyclopedia of Mathematics states that polynomial coefficients may belong, for example, to the rational, real, or complex numbers. The University of Kentucky notes that non-real roots are forced to occur in conjugate pairs only when the polynomial has real coefficients.
So for a general quartic there is no requirement that both roots in
[
a(x-l)^3(x-m)=0
]
be real. A direct counterexample is
[
(x-(1+i))^3(x-(2-3i))=0,
]
which is a quartic with a triple root at (1+i) and a simple root at (2-3i); neither root is real.
Therefore the quoted condition is incorrect unless the article explicitly restricts this classification to quartics with real coefficients.
2 sources
- Polynomial - Encyclopedia of Mathematics
With regard to the coefficients of a polynomial one assumes that they belong to a field, for example, the field of rational, real or complex numbers.
- MA 109 College Algebra Chapter 3 - University of Kentucky
If f(x) is a non-zero polynomial with real coefficients, then the non-real complex roots of f(x) = 0 occur in complex conjugate pairs.
where l and m are two different real numbers or a pair of non-real complex conjugate numbers.
A quartic with two double roots does not need those roots to be real or conjugate. That restriction only applies when coefficients are real, but the article states it as part of the general case.
Full reasoning
This claim is too restrictive for a general quartic.
A polynomial may have complex coefficients; the Encyclopedia of Mathematics explicitly allows coefficients from the rational, real, or complex numbers. The University of Kentucky notes that non-real roots must come in conjugate pairs only when the polynomial has real coefficients.
Without that real-coefficient assumption, the double roots in
[
a(x-l)^2(x-m)^2=0
]
need not be real or complex conjugates. For example,
[
(x-(1+i))^2(x-2i)^2=0
]
has two distinct double roots, (1+i) and (2i), which are neither both real nor a conjugate pair.
So the quoted classification is incorrect as written for the general quartic case; it would need an explicit restriction to real coefficients.
2 sources
- Polynomial - Encyclopedia of Mathematics
With regard to the coefficients of a polynomial one assumes that they belong to a field, for example, the field of rational, real or complex numbers.
- MA 109 College Algebra Chapter 3 - University of Kentucky
If f(x) is a non-zero polynomial with real coefficients, then the non-real complex roots of f(x) = 0 occur in complex conjugate pairs.
where l , m , and n are three different real numbers or l is a real number and m and n are a pair of non-real complex conjugate numbers.
This root description is not valid for all quartics. It is only guaranteed for real-coefficient quartics; a general quartic may instead have arbitrary complex roots.
Full reasoning
This statement incorrectly treats a real-coefficient restriction as though it applied to the general quartic case.
Polynomial coefficients may be complex as well as real. The Encyclopedia of Mathematics states that polynomial coefficients may belong, for example, to the rational, real, or complex numbers. The University of Kentucky notes that non-real roots are forced to appear in conjugate pairs only for polynomials with real coefficients.
So a quartic of the form
[
a(x-l)^2(x-m)(x-n)=0
]
does not, in general, need to have either three real roots or one real root plus a conjugate pair. For example,
[
(x-(1+i))^2(x-2i)(x-(3-i))=0
]
is a quartic with a double root at (1+i) and simple roots at (2i) and (3-i); this does not fit the article's stated alternatives.
Therefore the quoted claim is false as written unless the discussion is explicitly limited to quartics with real coefficients.
2 sources
- Polynomial - Encyclopedia of Mathematics
With regard to the coefficients of a polynomial one assumes that they belong to a field, for example, the field of rational, real or complex numbers.
- MA 109 College Algebra Chapter 3 - University of Kentucky
If f(x) is a non-zero polynomial with real coefficients, then the non-real complex roots of f(x) = 0 occur in complex conjugate pairs.