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  2. Rationalisation (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Rationalisation_(mathematics)

    In elementary algebra, root rationalisation (or rationalization) is a process by which radicals in the denominator of an algebraic fraction are eliminated.. If the denominator is a monomial in some radical, say , with k < n, rationalisation consists of multiplying the numerator and the denominator by , and replacing by x (this is allowed, as, by definition, a n th root of x is a number that ...

  3. Partial fraction decomposition - Wikipedia

    en.wikipedia.org/wiki/Partial_fraction_decomposition

    In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

  4. Conjugate (square roots) - Wikipedia

    en.wikipedia.org/wiki/Conjugate_(square_roots)

    As (+) = and (+) + =, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator , by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation ).

  5. Periodic continued fraction - Wikipedia

    en.wikipedia.org/wiki/Periodic_continued_fraction

    He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other.

  6. Algebraic number - Wikipedia

    en.wikipedia.org/wiki/Algebraic_number

    This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, tan ⁠ 3 π / 16 ⁠, tan ⁠ 7 π / 16 ⁠, tan ⁠ 11 π / 16 ⁠, and tan ⁠ 15 π / 16 ⁠ satisfy the irreducible polynomial x 4 − 4x 3 − 6x 2 + 4x + 1 = 0, and so are conjugate algebraic integers. This is the ...

  7. Conjugate element (field theory) - Wikipedia

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

    Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates. This follows as K ( α ) is K -isomorphic to K ( β ) by irreducibility of the minimal polynomial, and any isomorphism of fields F and F ' that maps polynomial p to p ' can be extended to an isomorphism of the splitting fields of p over F ...

  8. Algebraic integer - Wikipedia

    en.wikipedia.org/wiki/Algebraic_integer

    The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of and A is exactly . The rational number ⁠ a / b ⁠ is not an algebraic integer unless b divides a. The leading coefficient of the polynomial bx − a is the integer b.

  9. Rational root theorem - Wikipedia

    en.wikipedia.org/wiki/Rational_root_theorem

    If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.