Binomial equation

Binomial polynomial orbinomial expressionin () binomial) means that it is the sum of two terms (ie each term). The binomial expression is the simplest kind of polynomial following the monomial.

Definition

Since a binomial expression is a polynomial which is a sum of two, a binomial expression (one-sided binomial expression or () binomial expression) concerning one (or) x is replaced by an appropriate constant a, b And different m, n

axm - bxn {\ displaystyle ax ^ {m} - bx ^ {n}}! [{\ \ displaystyle ax ^ {m} - bx ^ {n}}] (https://wikimedia.org/api/rest_v1/ media / math / render / svg / 38743c5eefae45ffc0bfbe105873e8ba254846fc)

You can write in the form of. In the context of thinking, the Laurent binomial expression (or simply binomial expression) is the same as the previous expression on the shape, but the exponents m, n Is allowed to be a negative integer.

More generally, multivariate binomials are

ax 1 n 1 ⋯ xini - bx 1 m 1 ⋯ xjmj {\ displaystyle ax_ {1} ^ {n_ {1}} \ dotsb x_ {i} ^ {n_ {i}} - bx_ {1} ^ {m_ {1 } \ dotsb x_ {j} ^ {m_ {j}}}! [{\ \ displaystyle ax_ {1} ^ {n_ {1}} \ dotsb x_ {i} ^ {n_ {i}} - bx_ { 1} ^ {m_ {1}} \ dotsb x_ {j} ^ {m_ {j}}}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/e95c6001b5ffaa51be41e24b998efe5fc5aa68f3)

You can write in the form of. For example

3 x - 2 x 2 {\ displaystyle 3x - 2 x ^ {2}}! [{\ displaystyle 3x - 2 x ^ {2}}] (https://wikimedia.org/api/rest_v1/media/math/render / svg / 7645905 d 737 f 0 36 f 66 c 237 e 1 c 3 a 8 e 496 62 e 620 c 1) xy + yx 2 {\ displaystyle xy + yx ^ {2}}! [{\ \ displaystyle xy + yx ^ {2}}] (https://wikimedia.org/api/rest_v1/media/math/render/svg / 434 bb d 5 d 2 d 48 11 fb 9 ac 63 fcda 011 eb 4 cc f 7 d 5 ff) 0.9 x 3 + π y 2 {\ displaystyle 0.9 x ^ {3} + \ pi y ^ {2}}! {{\ Displaystyle 0.9 x ^ {3} + \ pi y ^ {2}}] (https : //wikimedia.org/api/rest_v1/media/math/render/svg/8d7b3987d10e1e15f22666b83179d24258ca5cf4)

Are binomial expressions.

Operations on simple binomial expressions

• The binomial expression x _ 2 - _ y _ 2 is added to the product of two binomial expressions: _ x _ 2 - _ y _ 2 = ( x _ \ + _ y ) ( x _ - _ y _).
• More generally, x _ n _ + 1 - _ y _ n _ + 1 = ( x _ - _ y ) Σ _ n _k - = 0 _ x kyn - k - holds.

• If you are considering polynomials of coefficients, you can also consider x 2 \ + _ y _ 2 = _ x _ 2 - ( iy ) 2 = ( x _ - _ iy ) ( x _ \ _ _ _ _ y) as another generalization.

• Product (ax _ \ + _ b ) ( cx _ \ + _ d ) = _ acx _ 2 \ + ( ad _ \ + _ bc ) _ x _ \ + _ bd _ of two primary binomial expressions (ax _ \ + _ b ) and ( cx _ \ + _ d ) .
• Binomial exponentiation, that is, (x \ + _ y ) _ n _ of the binomial expression _ x _ \ + _ y _ can be developed according to the meaning (or the same thing). For example, the square of the binomial expression _ x _ \ + _ y _ is equal to the sum of the square of each term plus twice the product of terms of each other: (_x _ \ + _ y ) ^ 2 = _ x _ 2 \ + 2 _ x x _ \ + _y _ 2.
• The set of coefficients (1, 2, 1) of each term appearing in this expansion expression is, and appears in the second row from the top. Likewise, using the number appearing in the nth row, we can also calculate n - th power expansion.
• It can be applied to "(m , _ n _) - official" for generating the formula for the square of the above binomial equation:

a _ 2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2. * The binomial expression expressed in the sum or difference of two cubes can be factored into a lower order polynomial as follows:

x _ 3 \ + _ y _ 3 = ( x _ \ + _ y ) ( x _ 2 - _ x _ \ + _ y _ 2), x _ 3 - _ y _ 3 = ( x _ - _ y ) ( x _ 2 \ + _ x _ \ + _ y _ 2).

Related item

• () (Which contains a large number of related links)

note

1. ****.. (English) . CS 1 maint: Multiple names: authors list 2. _CBMS Regional Conference Series in Mathematics (Conference Board of the Mathematical Sciences) (97): 62. Browsed on March 21, 2014. .

References

• L. Bostock, and S. Chandler (1978). Pure Mathematics 1.. Pp. 36.

.. (English) . CS 1 maint: Multiple names: authors list Hazewinkel, Michiel, ed. (2001),, __,,,: (Note that binomial formulas are also called binomials)

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Post Date : 2018-02-03 05:00