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

例如，数家最近发现了一个展开这种多项式的法。

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所有的根。

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

多项式也可以用于近似对数和三角函数这些函数手算相当麻烦。

By the time I got to where I wanted it, it had like an eighth-degree polynomial in the denominator, which, under normal circumstances, would mean " No, you're not going to be able to do this."

当我解到我想要的答案时，它的分母中有八次多项式——在正常情况下，这意味着 " 不，这个题你不可能解来" 。

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。

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多项式简化成二次程，然后从二次程中得黄金分割率。

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

Charles Babbage提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。

You can do the exact same thing with code too, because code can behave kind of like those polynomials and you can simplify code without knowing what the underlying variables are even referring to.

你也可以用代码做同样的事情，因为代码的行为式有点像那些多项式， 你可以在不知道底层变量实际代表什么的情况下简化代码。

If you take the polynomial y cubed minus 2y equals 0, that factors as y minus square root of 2 times y plus square root of 2 times y minus 0.

如果你取多项式 y 立减去 2y 等于 0，它可以分解为 y 减去 根号2 乘以 y 加上 根号2 乘以 y 减去 0。

Luckily it turns out that there is a really simple way to test whether or not  one of these polynomials has an attracting cycle.

幸运的是，有一种非常简单的法来检验这些多项式是否具有吸引周期。

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式的综合除法课程，那你最好相信我会为此做好准备。

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，某个多项式，它是n的k次加上等等等等。

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

那个多项式的根只有虚数I和负眼。

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，可以是多项式或其他复杂的数函数。

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式的公式，我们称之为多项式项的独立项展开。

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式增长。

Imagine I tell you, I have some polynomial but I don't tell you what the coefficients are, those are a mystery to you.

想象一下， 我告诉你我有一个多项式， 但我没有告诉你系数是什么，这些对你来说是个谜。

However, some of you might complain that the only way that this is useful is  if we can actually evaluate this wild expression on our polynomial.

然而，你们中的一些人可能会抱怨，这种法只有在我们能够实际计算这个多项式的疯狂表达式的情况下才有用。

It turns out though, if you think about the group which permutes the roots of such a polynomial, there's something about the nature of this group that reveals that no quintic formula can exist.

事实证明， 如果你考虑一下置换这种多项式根的群，这个群的性质揭示了不存在五次程的求根公式。

Now what I want is some kind of way to visualize  every possible cubic polynomial at once with a single image in a way that shows which ones have  attracting cycles.

现在我想要的是某种法，能够一次性用一张图片可视化所有可能的三次多项式，并且显示哪些多项式具有吸引周期。