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

例如，数学最近发现了一个展开这种多项的方法。

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

Charles Babbage提出了一种称为差分机的新型机械装置这是一个可以近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。

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的多项。

And for the longest time mathematicians struggled to find a formula to solve degree 5 polynomials.

在很长一段时间里，数学在努力寻找求 5 次多项的公。

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

甚至还有一个 4 次多项的四次公，这绝对是一团糟。

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，它是多项 z 的第五次减一的根。

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

你所做的本质上是展开我写下的两个多项的完整乘积。

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

只需记住右侧的每个像素对应一个唯一的多项，由参数 lambda 确定。

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

让我的方程求器以同样的方继续，它最终收敛于该多项的许多零点。

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， 它在相关多项的根处有一个固定点。

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， 只需计算我已知的多项之一的所有这些样本，就已经进行了 N 平方运算。

Namely, taking a  discrete sequence that we want to understand and treating it as the coefficients on a polynomial, then evaluating that polynomial on complex values.

也就是说， 采用我想要理的离散序列并将其视为多项的系数，然后根据复数值评估该多项。

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.

当且仅当相应多项的牛顿法过程产生吸引循环时，此参数空间中的点才被着色为黑色。

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 由于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。

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，粗略的算法概述是， 每当您想要对两个数字列表进行卷积时，您将它视为两个多项的系数。

Which in the back of your mind, you can just think of as treating them like they're the coefficients of a polynomial and evaluating it at a very specially selected set of points.

在你的内心深处，你可以将它视为多项的系数，并在一组非常特别选择的点上对其进行评估。

We then constructed this polynomial whose coefficients  tell us how many subsets have a particular sum for each value n. So what we want is to add up every  fifth coefficient of that polynomial.

然后，我构造了这个多项，其系数告诉我有多少子集对于每个值 n 具有特定的总和。所以我想要的是将该多项的每五个系数相加。

But the actual way that it breaks down involves a different kind of simple group, a different kind of atom, one which polynomial solutions built up from radicals would never allow.

但它分的实际方涉及一种不同类型的简单群，一种不同类型的原子， 这是由根建立的多项所不允许的。