In other words, a "zero" of a function is an input value that produces an output of zero (0).[1]A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree and that the number of roots and the degree are equal when one considers the complex roots (or more generally the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by
has the two roots 2 and 3, since
If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept.
by regrouping all terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function f. In other words, "zero of a function" is a phrase denoting a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.
Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because one is the smallest odd whole number), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero in the process of changing from negative to positive or vice versa.
The fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs.[1]Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.『虛』、『實』分殊言『解析』，『求根』理則道之深︰
![︰ Zero of a function From Wikipedia, the free encyclopedia (Redirected from Root of a function) A graph of the function cos(x) on the domain , with x-intercepts indicated in red. The function has zeroes where x is , , and . In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x) = 0. In other words, a "zero" of a function is an input value that produces an output of zero (0).[1] A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree and that the number of roots and the degree are equal when one considers the complex roots (or more generally the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by has the two roots 2 and 3, since If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept. Solution of an equation Every equation in the unknown x may be rewritten as f(x) = 0 by regrouping all terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function f. In other words, "zero of a function" is a phrase denoting a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations. Polynomial roots Main article: Properties of polynomial roots Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because one is the smallest odd whole number), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero in the process of changing from negative to positive or vice versa. Fundamental theorem of algebra Main article: Fundamental theorem of algebra The fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs.[1] Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. 『虛』、『實』分殊言『解析』，『求根』理則道之深︰ 代數基本定理 代數基本定理說明，任何一個一元複係數方程式都至少有一個複數根。也就是說，複數域是代數封閉的。 有時這個定理表述為：任何一個非零的一元n次複係數多項式，都正好有n個複數根。這似乎是一個更強的命題，但實際上是「至少有一個根」的直接結果，因為不斷把多項式除以它的線性因子，即可從有一個根推出有n個根。 儘管這個定理被命名為「代數基本定理」，但它還沒有純粹的代數證明，許多數學家都相信這種證明不存在。[1]另外，它也不是最基本的代數定理；因為在那個時候，代數基本上就是關於解實係數或複係數多項式方程，所以才被命名為代數基本定理。 高斯一生總共對這個定理給出了四個證明，其中第一個是在他22歲時（1799年）的博士論文中給出的。高斯給出的證明既有幾何的，也有函數的，還有積分的方法。高斯關於這一命題的證明方法是去證明其根的存在性，開創了關於研究存在性命題的新途徑。 同時，高次代數方程的求解仍然是一大難題。伽羅瓦理論指出，對於一般五次以上的方程，不存在一般的代數解。 白努利數有其原，既不在分子](https://www.freesandal.org/wp-content/ql-cache/quicklatex.com-9831e4e0f966ea4e27b5d111ce098874_l3.png)
![\scriptstyle {[-2\pi ,2\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e03bf1e49393bd0adf23e73fc71a0256ea9183)
, with x-intercepts indicated in red. The function has zeroes where x is 
, 
, 
and 
. 
