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In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the ...
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function ...
Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
More concretely, an n-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K: (, …,) = = =,. This formula may be rewritten using matrices: let x be the column vector with components x 1 , ..., x n and A = ( a ij ) be the n × n matrix over K whose entries are the coefficients of q .
This restricts the possible terms in the discriminant. For the general quadratic polynomial, the discriminant is a homogeneous polynomial of degre 2 which has only two there are only two terms, while the general homogeneous polynomial of degree two in three variables has 6 terms. The discriminant of the general cubic polynomial is a homogeneous ...
For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term 2x in x 2 + 2x + 1 is a linear term in a quadratic polynomial.
A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form. where a ≠ 0. [1] The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or ...