polynomials 是什麼?

f.5 amaths 課程的polynomials(多項式) 有binomial(二項式) and trinomial(三項式),

for example, (5x + 1)^3 這是一個binomial
大多數會叫你拆開呢舊binomial
(5x+1)^3
=(5x)^3 + (3)(5x)^2 +(3)(5x) + 1
=125x^3+75x^2 + 15x +1

如果他問要x^2的coefficient(係數)
就是x^2 旁邊的數字,即是75
如果他問constant(常數)
就是整個binomal的純數字(沒有x),即是 1

binomial 公式是
(a + b)^n
=nC0 (a)^n (b)^0+ nC1 (a)^(n-1) (b)^1 +nC2 (a)^(n-2) (b)^2 +......+nCn (a)^0 (b)^n
如果是trinomial
(a+b+c)^n
就要把(a+b)弄做一項,以2項式計算
=nC0 (a+b)^n (c)^0+ nC1 (a+b)^(n-1) (c)^1 +nC2 (a+b)^(n-2) (c)^2 +....
..+nCn (a+b)^0 (c)^n

之後再拆入面a+b 個二項式,very麻煩

而polynomial 會有埋nCr 就係個d nC0 nC1,
nCr 公式係nCr =n!/[(r!)(n-r)!]

n!= (n)(n-1)(n-2)(n-3).......(1)

明未-.-

2007-06-30 04:29:56 補充：
polynomial多項式其實就是有很多不同項目有x^2有x有x^3有y有z有constant,而這些項目全加起來

In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power).
Thus,
2x^2yz^3 - 3y^3 + 5yz - 2

is a polynomial, but
1 / (x^2+1)

is not a polynomial.

A polynomial function is a function defined by evaluating a polynomial. For example, the function f defined by f(x) = x3−x is a polynomial function. Polynomial functions are an important class of smooth functions; smooth meaning that they are infinitely differentiable, i.e., they have derivatives of all finite orders.

Because of their simple structure, polynomials are easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.

polynomial係多項式
one example: (2y ﹢3x)(4y ﹢5x)
second example:2+2x+5x
coefficient(係數)of x = 7
constant:2

polynomials 是多項式