Is A Binomial A Polynomial

In mathematics, a polynomial with two terms

In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial.^[ane] It is the simplest kind of sparse polynomial after the monomials.

Definition [edit]

A binomial is a polynomial which is the sum of 2 monomials. A binomial in a single indeterminate (likewise known every bit a univariate binomial) tin can be written in the course

${\displaystyle ax^{m}-bx^{n}\,,}$

where a and b are numbers, and k and n are distinct nonnegative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a Laurent binomial, often merely called a binomial, is similarly defined, just the exponents yard and n may be negative.

More generally, a binomial may be written^[ii] as:

${\displaystyle ax_{1}^{n_{1}}\dotsb x_{i}^{n_{i}}-bx_{1}^{m_{1}}\dotsb x_{i}^{m_{i}}}$

Examples [edit]

${\displaystyle 3x-2x^{2}}$

${\displaystyle xy+yx^{ii}}$

${\displaystyle 0.9x^{3}+\pi y^{2}}$

${\displaystyle 2x^{3}+seven}$

Operations on unproblematic binomials [edit]

• The binomial x ^two − y ² can be factored as the product of ii other binomials:

${\displaystyle x^{2}-y^{2}=(ten-y)(x+y).}$

This is a special instance of the more than general formula:

${\displaystyle x^{n+1}-y^{north+i}=(x-y)\sum _{m=0}^{n}x^{k}\,y^{n-k}.}$

When working over the complex numbers, this tin also be extended to:

${\displaystyle x^{2}+y^{ii}=x^{2}-(iy)^{2}=(x-iy)(x+iy).}$

• The product of a pair of linear binomials (ax + b) and (cx + d) is a trinomial:

${\displaystyle (ax+b)(cx+d)=acx^{2}+(advertising+bc)ten+bd.}$

• A binomial raised to the n ^th power, represented as (x + y) ⁿ can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square (ten + y)² of the binomial (x + y) is equal to the sum of the squares of the two terms and twice the product of the terms, that is:

${\displaystyle (10+y)^{2}=ten^{two}+2xy+y^{2}.}$

The numbers (1, two, 1) appearing as multipliers for the terms in this expansion are binomial coefficients two rows down from the top of Pascal'due south triangle. The expansion of the due north ^th power uses the numbers n rows down from the top of the triangle.

• An application of above formula for the foursquare of a binomial is the "(m, n)-formula" for generating Pythagorean triples:

For m < n , permit a = n ² − m ² , b = iimn , and c = northward ^two + thousand ² ; and so a ² + b ² = c ² .

• Binomials that are sums or differences of cubes can be factored into lower-club polynomials equally follows:

${\displaystyle x^{3}+y^{3}=(x+y)(ten^{ii}-xy+y^{2})}$

${\displaystyle x^{3}-y^{3}=(ten-y)(ten^{2}+xy+y^{2})}$

See also [edit]

• Completing the square
• Binomial distribution
• List of factorial and binomial topics (which contains a large number of related links)

Notes [edit]

1. ^ Weisstein, Eric. "Binomial". Wolfram MathWorld. Retrieved 29 March 2011.
2. ^ Sturmfels, Bernd (2002). Solving Systems of Polynomial Equations. CBMS Regional Conference Series in Mathematics. Vol. 97. American Mathematical Society. p. 62. ISBN9780821889411.

References [edit]

• Bostock, L.; Chandler, South. (1978). Pure Mathematics 1. Oxford University Printing. p. 36. ISBN0-85950-092-half-dozen.

Is A Binomial A Polynomial,

Source: https://en.wikipedia.org/wiki/Binomial_(polynomial)

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