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The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial
Moreover, if one sets x = 1 + t, one gets without computation that () = (+) is a polynomial in t with the same first coefficient 3 and constant term 1. [2] The rational root theorem implies thus that a rational root of Q must belong to { ± 1 , ± 1 3 } , {\textstyle \{\pm 1,\pm {\frac {1}{3}}\},} and thus that the rational roots of P satisfy x ...
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/ x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a / b is b / a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one ...
A simple fraction (also known as a common fraction or vulgar fraction, where vulgar is Latin for "common") is a rational number written as a / b or , where a and b are both integers. [9] As with other fractions, the denominator (b) cannot be zero. Examples include 1 2 , − 8 5 , −8 5 , and 8 −5 .
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator p and a non-zero denominator q. [1] For example, is a rational number, as is every integer (e.g., ). The set of all rational numbers, also referred to as " the rationals ", [2] the field of rationals[3] or ...
The following is a rough outline of a proof. [1] First, some preliminary definitions: Write the polynomial f ( x ) {\displaystyle f(x)} as ∑ i = 0 n a i x b i {\displaystyle \sum _{i=0}^{n}a_{i}x^{b_{i}}} where we have integer powers 0 ≤ b 0 < b 1 < ⋯ < b n {\displaystyle 0\leq b_{0}<b_{1}<\cdots <b_{n}} , and nonzero coefficients a i ≠ ...
The reverse process of expressing a proper rational fraction as the sum of two or more fractions is called resolving it into partial fractions. For example, For example, 2 x x 2 − 1 = 1 x − 1 + 1 x + 1 . {\displaystyle {\frac {2x}{x^{2}-1}}={\frac {1}{x-1}}+{\frac {1}{x+1}}.}
Algebraic fraction. In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions. A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials.
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