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The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, f(x) = 0, is the x-axis. In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero ...
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
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.Two definitions of a monomial may be encountered: A monomial, also called a power product or primitive monomial, [1] is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. [2]
The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the canonical form, normal form, or expanded form of the polynomial. Given a polynomial expression, one can compute the expanded form of the represented polynomial by expanding with the distributive law all the ...
A polynomial equation over the rationals can always be converted to an equivalent one in which the coefficients are integers. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation y 4 + x y 2 = x 3 3 − x y 2 + y 2 − 1 7 {\displaystyle y^{4}+{\frac {xy}{2 ...
For example the above polynomial expression is equivalent (denote the same polynomial as + + Many author do not distinguish polynomials and polynomial expressions. In this case the expression of a polynomial expression as a linear combination is called the canonical form , normal form , or expanded form of the polynomial.
Given a quadratic polynomial of the form + + it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example: + + = [+ +] = [(+) +] = (+) + = (+) + This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms. The integer at the ...
For example, the 4th order Chebyshev polynomial from the example above is +, which by inspection contains no roots of zero. Creating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of X 4 − .828427 X 2 {\displaystyle X^{4}-.828427X^{2}} , which by inspection contains two ...