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  2. Quintic function - Wikipedia

    en.wikipedia.org/wiki/Quintic_function

    An example of a more complicated (although small enough to be written here) solution is the unique real root of x 5 − 5x + 12 = 0. Let a = √ 2φ −1, b = √ 2φ, and c = 4 √ 5, where φ = ⁠ 1+ √ 5 / 2 ⁠ is the golden ratio. Then the only real solution x = −1.84208... is given by

  3. Degree of a polynomial - Wikipedia

    en.wikipedia.org/wiki/Degree_of_a_polynomial

    The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term.

  4. Abel–Ruffini theorem - Wikipedia

    en.wikipedia.org/wiki/Abel–Ruffini_theorem

    [5] However, the existence of specific equations that cannot be solved in radicals seems to be a consequence of Abel's proof, as the proof uses the fact that some polynomials in the coefficients are not the zero polynomial, and, given a finite number of polynomials, there are values of the variables at which none of the polynomials takes the ...

  5. Polynomial - Wikipedia

    en.wikipedia.org/wiki/Polynomial

    Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞). [10] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots.

  6. Homogeneous polynomial - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_polynomial

    In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. [1] For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.

  7. Fundamental theorem of algebra - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

    The numerator of the rational expression being integrated has degree at most n − 1 and the degree of the denominator is n + 1. Therefore, the number above tends to 0 as r → +∞. But the number is also equal to N − n and so N = n. Another complex-analytic proof can be given by combining linear algebra with the Cauchy theorem.

  8. Minimal polynomial (field theory) - Wikipedia

    en.wikipedia.org/wiki/Minimal_polynomial_(field...

    The minimal polynomial f of α is unique.. To prove this, suppose that f and g are monic polynomials in J α of minimal degree n > 0. We have that r := f−g ∈ J α (because the latter is closed under addition/subtraction) and that m := deg(r) < n (because the polynomials are monic of the same degree).

  9. Cyclotomic polynomial - Wikipedia

    en.wikipedia.org/wiki/Cyclotomic_polynomial

    The case of the 105th cyclotomic polynomial is interesting because 105 is the least positive integer that is the product of three distinct odd prime numbers (3×5×7) and this polynomial is the first one that has a coefficient other than 1, 0, or −1: [3]