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

例如，数近发现了一个展开这种项方法。

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新型机械装置这是一个可以近似项更加复杂机器。

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

项也可以用于近似对数和三角函数这些函数手算相当麻烦。

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

由于项可以有任意大阶 所以这组基函数是无限。

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

项描述了几个变量之间关系 例如射程和大气压力，或者Carrie Anne要吃少披萨才会开心。

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；它们可以是项或其他一些奇特数函数。

For example, let's say I gave you two really big polynomials.

例如，假设我给了你两个非常大项。

What do polynomials have  to do with things?

项与事物有什么关系？

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

那个项只有虚数I和负眼。

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，这个项有很对称性，我可以利用这种对称性推导出所有。

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项看做向量时 它们有无穷个坐标。

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项上 比如X²+3 x+5 4X。

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.

我可以把我答案从八次方项简化成二次方程，然后从二次方程中得出黄金分割率。

Another way to think about that is  that they are roots of the polynomial z to the fifth minus one.

另一种思考方是，它们是项 z 第五次减一。

And, in this case, it's an infinite polynomial so  I should really be calling it a power series.

而且，在这种情况下，它是一个无限项，所以我真应该将其称为幂级数。

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

这将是这个大项——第 k 次项。

Because for one thing, just calculating all these samples for one of the polynomials we know, already takes on the order of N squared operations.

因为一方面， 只需计算我们已知项之一所有这些样本，就已经进行了 N 平方运算。

In the case of the functions arising from Newton's method, by  design they have a fixed point at the roots of the relevant polynomial.

对于由牛顿法产生函数，据设计， 它们在相关项处有一个固定点。

One context where this view is especially natural, is when you multiply together two polynomials.

这种观点特别自然一种情况是当您将两个项相乘时。

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 由于每个单独项只有有限个项 它坐标将是一个有限数字串和一个无限零故事。