For example, mathematicians recently discovered a way to expand it to these kinds of polynomials.

例如，数学家最发现了一个展开这种多项的方法。

In algebra, his work expanded our understanding of polynomial equations, opening new doors of exploration in complex numbers.

在代数方面，他的工作扩展了我们对多项方程的理解，打开了探索复数的新大门。

Polynomials could also be used to approximate logarithmic and trigonometric functions, which are a real hassle to calculate by hand.

多项也可以似对数和三角函数这些函数手算相当麻烦。

Charles Babbage proposed a new mechanical device called the Difference Engine, a much more complex machine that could approximate polynomials.

Charles Babbage提出了一种称为差分机的新型机械装置这是一个可以似多项的更加复杂的机器。

The only roots of that polynomial are the imaginary numbers I and negative eye.

那个多项的根只有虚数I和负眼。

Polynomials describe the relationship between several variables - like range and air pressure, or amount of pizza Carrie Anne eats and happiness.

多项描述了几个变量之间的关系 例如射程和大气压，Carrie Anne要吃多少披萨才会开心。

And these don’t have to be straight lines; they can be polynomials or some other fancy mathematical function.

这些不一定是直线；它们可以是多项其他一些奇特的数学函数。

Let's limit ourselves to polynomials, things like X squared plus three, X plus five or four x.

让我们把自己限制在多项上 比如X²+3 x+5 4X。

Since our polynomials can have arbitrarily large degree, this set of basis functions is infinite.

由多项可以有任意大的阶 所以这组基函数是无限的。

But it turned out that the polynomial had a lot of symmetry and I could then exploit that symmetry to deduce all the roots.

但事实证明，这个多项有很多对称性，我可以利这种对称性推导出所有的根。

It's going to be this big polynomial-- kth degree polynomial.

这将是这个大多项——第 k 次多项。

Algebra In the field of algebra, Gauss proved the fundamental theorem of algebra, which states that every non-constant polynomial equation has at least one complex root.

代数 在代数领域，高斯证明了代数基本定理，即每个非常数多项方程至少有一个复根。

But that's okay. It just means that when we treat our polynomials as vectors, they're going to have infinitely many coordinates.

不过没关系 它只是意味着当我们把多项看做向量时 它们有无穷多个坐标。

I was able to reduce what I had to find from an eighth-degree polynomial to a quadratic, and from the quadratic, the golden ratio fell out.

我可以把我的答案从八次方多项简化成二次方程，然后从二次方程中得出黄金分割率。

Each of the polynomials in our space will only have finitely many terms, but the full space is going to include polynomials with arbitrarily large degrees.

空间中的每一个多项都只有有限项 但是整个空间包含了任意大阶的多项。

In general, since every individual polynomial has only finitely many terms, its coordinates will be some finite string of numbers with an infinite tale of zeroes.

一般来说 由每个单独的多项只有有限个项 它的坐标将是一个有限的数字串和一个无限的零故事。

Take the coordinates representing the polynomial X cubed plus five, X squared, plus four, X plus five, then put those coordinates on the right of the matrix.

取表示多项(X³+5)(X²+4)(X+5)的坐标 然后把这些坐标放到矩阵的右边。

Since polynomials are already written down as the sum of scaled powers of the variable X, it's pretty natural to just choose pure powers of X as the basis function.

由多项已经被写成变量X的倍数幂之和 选择X的纯幂作为基函数是很自然的。

By the time I got to where I wanted it, it had like an eighth-degree polynomial in the denominator, which, under normal circumstances, would mean " No, you're not going to be able to do this."

当我解到我想要的答案时，它的分母中有八次方多项——在正常情况下，这意味着 " 不，这个题你不可能解出来" 。