jesh0975556
14.03.2020 •
Mathematics
Combine the like terms to create an equivalent expression:
{-4p+(-6p)}
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Ответ:
Answer and Step-by-step explanation:
Ответ:
Step-by-step explanation:
Ответ:
A simple polynomial identity is:
x2−y2=(x−y)(x+y)
In fact this is just:
x2–1=(x−1)(x+1)
in disguise.
The benefit of such an identity is that it enables us to quickly find solutions when we equate the identity to 0 . In my example above this benefit is not easy to spot, we could use either half of the given identities and set it to 0 . But this is not always the case. For instance:
x2–2xy+y2=0
is already more difficult to solve than:
(x−y)(x−y)=0
Splitting a polynomial into factors, like in the examples above will give us an identity. The amount of factors we may find depend on the type of numbers we allow x,y,⋯ to represent.
For instance you might have a hard time splitting:
x2+1
if we use x∈R .
But if we allow x∈C we can factor this polynomial!
x2+1=(x+i)(x−i)
This also means that x2+1=0 has two solutions: x=±i . You may check this identity by using i2=−1 .
In fact, a polynomial in one variable can always be split into polynomial factors of degree 1 . This is called the fundamental theorem of algebra. The extension of R to C enables us to factor a polynomial in one variable of degree n into exactly n factors, or which is the same thing, enables us to express all roots in the ‘field’ we use.
This difference, between being able to split a polynomial completely or only partly, depending on what values we allow for x,y,⋯ does exist in a broader context. You will meet extensions, like from R→C , in many varieties and truly appreciate what the word algebra means.
Step-by-step explanation: