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方法。

In algebra, his work expanded our understanding of polynomial equations, opening new doors of exploration in complex numbers.

在代数方面，他工作扩展了我们对多方程理解，打开了探索复数新大门。

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

多也可以用于近似对数和三角函数这些函数手算相当麻烦。

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新型机械装置这是一个可以近似多更加复杂机器。

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

那个多根只有虚数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.

这些不一定是直线；它们可以是多或其他一些奇特数学函数。

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

让我们把制在多上 比如X²+3 x+5 4X。

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

由于多可以有任意大阶 所以这组基函数是无。

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，这个多有很多对称性，我可以利用这种对称性推导出所有根。

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

这将是这个大多——第 k 次多。

Algebra In the field of algebra, Gauss proved the fundamental theorem of algebra, which states that every non-constant polynomial equation has at least one complex root.

代数 在代数领域，高斯证明了代数基本定理，即每个非常数多方程至少有一个复根。

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看做向量时 它们有无穷多个坐标。

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简化成二次方程，然后从二次方程中得出黄金分割率。

Each of the polynomials in our space will only have finitely many terms, but the full space is going to include polynomials with arbitrarily large degrees.

空间中每一个多都只有有 但是整个空间包含了任意大阶多。

In general, since every individual polynomial has only finitely many terms, its coordinates will be some finite string of numbers with an infinite tale of zeroes.

一般来说 由于每个单独多只有有个 它坐标将是一个有数字串和一个无零故事。

Take the coordinates representing the polynomial X cubed plus five, X squared, plus four, X plus five, then put those coordinates on the right of the matrix.

取表示多(X³+5)(X²+4)(X+5)坐标 然后把这些坐标放到矩阵右边。

Since polynomials are already written down as the sum of scaled powers of the variable X, it's pretty natural to just choose pure powers of X as the basis function.

由于多已经被写成变量X倍数幂之和 选择X纯幂作为基函数是很然。

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."

当我解到我想要答案时，它分母中有八次方多——在正常情况下，这意味着 " 不，这个题你不可能解出来" 。