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

例，数学家最近发现了一个展开这种多式的方法。

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一个可以近似多式的更加复杂的机器。

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

多式也可以用于近似对数和三角函数这些函数手算相当麻烦。

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

例，我给了你两个非常大的多式。

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

由于多式可以有任意大的阶 所以这组基函数的。

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

多式描述了几个变量之间的关系 例射程和大气压力，或者Carrie Anne要吃多少披萨才会开心。

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

这些不一定直线；它们可以多式或其他一些奇特的数学函数。

What do polynomials have  to do with things?

多式与事物有什么关系？

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

那个多式的根只有虚数I和负眼。

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 它只意味着当我们把多式看做向量时 它们有穷多个坐标。

Let's limit ourselves to polynomials, things like X squared plus three, X plus five or four x.

让我们把自己制在多式上 比X²+3 x+5 4X。

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，这个多式有很多对称性，我可以利用这种对称性推导出所有的根。

It's going to be this big polynomial-- kth degree polynomial.

这将这个大多式——第 k 次多式。

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多式简化成二次方程，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.

另一种思考方式，它们多式 z 的第五次减一的根。

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

而且，在这种情况下，它一个多式，所以我真的应该将其称为幂级数。

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

这种观点特别自然的一种情况当您将两个多式相乘时。

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解 5 次多式的公式。

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

甚至还有一个 4 次多式的四次公式，这绝对一团糟。

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， 只需计算我们已知的多式之一的所有这些样本，就已经进行了 N 平方运算。