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

例如，数学家最近发现个展开这的方法。

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

Charles Babbage提出称为差分机的新型机械装置这是个可以近似的更加复杂的机器。

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

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.

由于可以有任意大的阶 所以这组基函数是无限的。

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

描述几个变量之间的关系 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.

这些不定是直线；它们可以是或其他些奇特的数学函数。

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

例如，假设我给你两个非常大的。

What do polynomials have  to do with things?

与事物有什么关系？

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

那个的根只有虚数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，这个有很对称性，我可以利用这对称性推导出所有的根。

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.

我可以把我的答案从八次方简化成二次方程，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.

另思考方是，它们是 z 的第五次减的根。

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

而且，在这情况下，它是个无限，所以我真的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——第 k 次。

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.

因为方面， 只需计算我们已知的之的所有这些样本，就已经进行 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.

对于由牛顿法产生的函数，根据设计， 它们在相关的根处有个固定点。

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

这观点特别自然的情况是当您将两个相乘时。

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.

般来说 由于每个单独的只有有限个 它的坐标将是个有限的数字串和个无限的零故事。