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多项式更加复杂机器。

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

例如，假设我给了你两个非多项式。

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。

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

在很长一段时间里，数学家们都在努力寻找求 5 次多项式公式。

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

另一种思考方式是，它们是多项式 z 第五次减一根。

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

甚有一个 4 次多项式四次公式，这绝对是一团糟。

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。

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

你所做本质上是展开我写下两个多项式完整乘积。

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， 只需计算我们已知多项式之一所有这些样本，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，根据设计， 它们在相关多项式根处有一个固定点。

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

让我们方程求器以同样方式继续，它最终收敛于该多项式许多零点。

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， 采用我们想要理离散序列并将其视为多项式系数，然后根据复数值评估该多项式。

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。

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阶多项式。

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.

当且仅当相应多项式牛顿法过程产生吸引循环时，此参数空间中点才被着色为黑色。

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，您都将它们视为两个多项式系数。

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.

在你内心深处，你可以将它们视为多项式系数，并在一组非特别选择点上对其进行评估。

We have a question that's just about subsets, it's a discrete math  problem, and yet the way that we can answer it is to evaluate a crazy polynomial on some judiciously  chosen complex numbers.

我们有一个关于子集问题，这是一个离散数学问题， 但我们回答它方法是在一些明智选择复数上评估一个疯狂多项式。

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.

但它分实际方式涉及一种不同类型简单群，一种不同类型原子， 这是由根式建立多项式所不允许。