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

例如，数学家最近发现了一个展开这种多项式法。

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多项式更加复杂机器。

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

那个多项式根只有虚数I和负眼。

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

例如，假设我给你两个非常大多项式。

A formula which involves an expansion in separate terms, polynomial terms as we say, for certain functions.

一个涉及特定函数展开式公式，我称之为多项式项独立项展开。

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

而且这些不一定是直线，可以是多项式或其他复杂数学函数。

What you're doing is essentially expanding out the full product of the two polynomials I have written down.

你所做其实就是将我写下两个多项式完整乘积展开。

There's even a quartic formula for degree 4 polynomial, which is an absolute mess.

甚至有解四次多项式公式，那简直是一团糟。

And letting our equation solver continue in this same way, it eventually converges on lots of zeros of this polynomial.

让我程求解器继续以这种式运行，它最终会收敛到这个多项式许多零点。

And if I had to teach a lesson on synthetic division on fourth order polynomials, you better bet I would prepare for it.

而且如果我不得不教授关于四次多项式综合除法课程，那你最好相信我会为此做好准备。

Just remember each pixel on the right  corresponds to a unique polynomial, as determined by this parameter lambda.

只需记住右侧每个像素对应一个唯一多项式，由参数 lambda 确定。

This very specific context where convolutions show up, multiplying two polynomials, opens the doors for an algorithm that's relevant everywhere else where convolutions might come up.

这种非常特定卷积出现场景，即两个多项式乘法， 为一种算法打开了大门， 这种算法在其他可能出现卷积地都具有相关性。

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.

空间中每一个多项式都只有有限项 但是整个空间包含了任意大阶多项式。

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 由于每个单独多项式只有有限个项 它坐标将是一个有限数字串和一个无限零故事。

So the rough algorithm outline then would be whenever you want to convolve two lists of numbers, you treat them like they're coefficients of two polynomials.

因此，粗略算法概述是， 每当您想要对两个数字列表进行卷积时，您都将它视为两个多项式系数。

Points in this parameter space are colored black  if and only if the Newton's method process for the corresponding polynomial produces an attracting  cycle.

在这个参数空间中，如果且仅如果对应多项式牛顿法过程产生一个吸引周期，则该点被着色为黑色。

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

另一种思考式是，它是多项式z五次减一根。

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，它是一个无穷多项式级数，所以我应该称之为幂级数。

And then suppose I say, I'll just tell you what the outputs of this polynomial would be if you inputted various different inputs like 0, 1, 2, 3 on and on.

然后假设我说，我会告诉你这个多项式在输入不同值比如0, 1, 2, 3等等时输出结果。

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.

因为一面， 仅仅计算已知多项式所有样本值，就已经需要大约N平次操作。