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

例如，数学家最近一展开这种多项的方法。

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

Charles Babbage提出一种称为差分机的新型机械装置这是一可以近似多项的更加复杂的机器。

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.

在一特别自然的场景中，当你将两多项相乘时。

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

那多项的根只有虚数I和负眼。

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

多项描述几变量之间的关系 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数增长，但结果却它是多项增长。

A formula which involves an expansion in separate terms, polynomial terms as we say, for certain functions.

一涉及特定函数展开的公，称之为多项项的独立项展开。

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的数学函数。

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

让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.

你所做的其实就是将写下的两多项的完整乘积展开。

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

例如，假设给你两非常大的多项。

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的过程正好与卷积相同，这挺酷的。

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的阶 所以这组基函数是无限的。

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

例如，两输出就足以唯一确定一线性多项。

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，这多项有很多对称性，可以利用这种对称性推导出所有的根。

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，某多项，它是n的k次方加上等等等等。

Apolynomial, like X squared plus three, X plus five, E.G.

多项 比如X²+3X+5 例如。

And if I had to teach a lesson on synthetic division on fourth order polynomials, you better bet I would prepare for it.

而且如果不得不教授关于四次多项的综合除法课程，那你最好相信会为此做好准备。

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.

可以把的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。