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型械装置这是一个可以近似多项更加复杂器。

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

多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。

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

在一个特别自然场景中，当你将两个多项相乘时。

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

那个多项根只有虚数I和负眼。

For example, you may imagine that something is logarithmic, but it turns out that it's polynomial.

例如，你或许认为某事物遵循对数增长，但结果却发现它是多项增长。

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

多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要吃多才会开心。

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公，我们称之为多项项独立项展开。

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，可以是多项或其他复杂数学函数。

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多项上 比如X²+3 x+5 4X。

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完整乘积展开。

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。

Which is pretty neat, expanding a polynomial and collecting like terms is exactly the same process as a convolution.

将多项展开并合并同类项过程正好与卷积相同，这挺酷。

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

由于多项可以有任意大阶 所以这组基函数是无限。

For example, two outputs are enough to uniquely specify a linear polynomial.

例如，两个输出就足以唯一确定一个线性多项。

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，这个多项有很多对称性，我可以利用这种对称性推导出所有根。

The limit as n approaches infinity of some polynomial where it's n to the kth power plus blah, blah, blah, blah.

当n趋于无穷大时，某个多项，它是nk次方加上等等等等。

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综合除法课程，那你最好相信我会为此做好准备。

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多项简化成二次方程，然后从二次方程中得出黄金分割率。

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多项看做向量时 它们有无穷多个坐标。