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

例如，家最近发现了一个展开这种多项式的方法。

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是一个可以近似多项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.

多项式也可以用近似对和三角这些手算相当麻烦。

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式。

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

由多项式可以有任意大的阶 所以这组基是无限的。

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

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

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增长，但结果却发现它是多项式增长。

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

这些不一定是直线；它们可以是多项式或其他一些奇特的。

What do polynomials have  to do with things?

多项式与事物有什么关系？

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

那个多项式的根只有虚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 它只是意味着当我们把多项式看做向量时 它们有无穷多个坐标。

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

让我们把自己限制在多项式上 比如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，这个多项式有很多对称性，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.

这将是这个大多项式——第 k 次多项式。

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项式简化成二次方程，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.

另一种思考方式是，它们是多项式 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，它是一个无限多项式，所以我真的应该将其称为幂级。

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是当您将两个多项式相乘时。

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

在很长一段时间里，家们都在努力寻找求解 5 次多项式的公式。

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

甚至还有一个 4 次多项式的四次公式，这绝对是一团糟。