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The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law.The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra.
For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...
If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).
In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f ( x ) {\displaystyle f(x)} is a polynomial, then x − a {\displaystyle x-a} is a factor of f ( x ) {\displaystyle f(x)} if and only if f ( a ) = 0 {\displaystyle f(a)=0} (that is, a {\displaystyle a} is a root of the polynomial).
≡ 2.54 cm ≡ 1 ⁄ 36 yd ≡ 1 ⁄ 12 ft ≡ 0.0254 m league (land) lea ≈ 1 hour walk, Currently defined in US as 3 Statute miles, [3] but historically varied from 2 to 9 km ≈ 4828 m: light-day: ≡ 24 light-hours ≡ 2.590 206 837 12 × 10 13 m: light-hour: ≡ 60 light-minutes ≡ 1.079 252 8488 × 10 12 m: light-minute: ≡ 60 light ...
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong.
The choice of relaxation factor ω is not necessarily easy, and depends upon the properties of the coefficient matrix. In 1947, Ostrowski proved that if A {\displaystyle A} is symmetric and positive-definite then ρ ( L ω ) < 1 {\displaystyle \rho (L_{\omega })<1} for 0 < ω < 2 {\displaystyle 0<\omega <2} .
We remember that the Euclid’s formulas do not give all Pythagorean triples that involves a predetermined positive integer x, for example, the triples (12,9,15), (33,180,183), and (33,44,55). Moreover, it can be laborious to find m and n such that x = m 2 − n 2 .