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

例如，数学家最近发现了展开这种多项方法。

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项方程理解，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.

多项也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提出了种称为差分新型械装置这是以近似多项更加复杂。

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

那多项根只有虚数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.

多项描述了几变量之间关系 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.

这些不定是直线；它们以是多项或其他些奇特数学函数。

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

让我们把自己限制在多项上 比如X²+3 x+5 4X。

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

由于多项以有任意大阶 所以这组基函数是无限。

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，我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.

这将是这大多项——第 k 次多项。

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，即每非常数多项方程至少有复根。

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项看做向量时 它们有无穷多坐标。

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.

我以把我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。

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.

空间中每多项都只有有限项 但是整空间包含了任意大阶多项。

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.

般来说 由于每单独多项只有有限项 它坐标将是有限数字串和无限零故事。

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.

取表示多项(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.

由于多项已经被写成变量X倍数幂之和 选择X纯幂作为基函数是很自然。

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项——在正常情况下，这意味着 " 不，这题你不能解出来" 。