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The remaining terms provide the leading-order equation, or leading-order balance, [5] or dominant balance, [6] [7] [8] and creating a new equation just involving these terms is known as taking an equation to leading-order. The solutions to this new equation are called the leading-order solutions [9] [10] to the original equation.
The coefficient attached to the highest degree of the variable in a polynomial of one variable is referred to as the leading coefficient; for example, in the example expressions above, the leading coefficients are 2 and a, respectively.
Let () be a polynomial equation, where P is a univariate polynomial of degree n. If one divides all coefficients of P by its leading coefficient, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial. For example, the equation
Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution x = p ⁄ q, written in lowest terms so that p and q are relatively prime, satisfies: p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n.
There exists a general formula for finding the roots to quartic equations, provided the coefficient of the leading term is non-zero. However, since the general method is quite complex and susceptible to errors in execution, it is better to apply one of the special cases listed below if possible.
For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number p, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient ka k understood to mean the sum of k copies of a k.
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 any given n ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials): = is the one of which the maximal absolute value on the interval [−1, 1] is minimal. This maximal absolute value is: 1 2 n − 1 {\displaystyle {\frac {1}{2^{n-1}}}} and | f ( x ) | reaches this maximum exactly n + 1 times at: x = cos ...