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- Binomial (polynomial) - Wikipedia
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- Binomial - Definition, Operations on Binomials & Examples - BYJU'S
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In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials.
Definition
A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form
a
x
m
−
b
x
n
,
{\displaystyle ax^{m}-bx^{n},}
where a and b are numbers, and m and n are distinct non-negative 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 simply called a binomial, is similarly defined, but the exponents m and n may be negative.
More generally, a binomial may be written as:
a
x
1
n
1
⋯
x
i
n
i
−
b
x
1
m
1
⋯
x
i
m
i
{\displaystyle a\,x_{1}^{n_{1}}\dotsb x_{i}^{n_{i}}-b\,x_{1}^{m_{1}}\dotsb x_{i}^{m_{i}}}
Examples
3
x
−
2
x
2
{\displaystyle 3x-2x^{2}}
x
y
+
y
x
2
{\displaystyle xy+yx^{2}}
0.9
x
3
+
π
y
2
{\displaystyle 0.9x^{3}+\pi y^{2}}
2
x
3
+
7
{\displaystyle 2x^{3}+7}
Operations on simple binomials
The binomial x2 − y2, the difference of two squares, can be factored as the product of two other binomials:
x
2
−
y
2
=
(
x
−
y
)
(
x
+
y
)
.
{\displaystyle x^{2}-y^{2}=(x-y)(x+y).}
This is a special case of the more general formula:
x
n
+
1
−
y
n
+
1
=
(
x
−
y
)
∑
k
=
0
n
x
k
y
n
−
k
.
{\displaystyle x^{n+1}-y^{n+1}=(x-y)\sum _{k=0}^{n}x^{k}y^{n-k}.}
When working over the complex numbers, this can also be extended to:
x
2
+
y
2
=
x
2
−
(
i
y
)
2
=
(
x
−
i
y
)
(
x
+
i
y
)
.
{\displaystyle x^{2}+y^{2}=x^{2}-(iy)^{2}=(x-iy)(x+iy).}
The product of a pair of linear binomials (ax + b) and (cx + d ) is a trinomial:
(
a
x
+
b
)
(
c
x
+
d
)
=
a
c
x
2
+
(
a
d
+
b
c
)
x
+
b
d
.
{\displaystyle (ax+b)(cx+d)=acx^{2}+(ad+bc)x+bd.}
A binomial raised to the nth 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 (x + 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:
(
x
+
y
)
2
=
x
2
+
2
x
y
+
y
2
.
{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}
The numbers (1, 2, 1) appearing as multipliers for the terms in this expansion are the binomial coefficients two rows down from the top of Pascal's triangle. The expansion of the nth power uses the numbers n rows down from the top of the triangle.
An application of the above formula for the square of a binomial is the "(m, n)-formula" for generating Pythagorean triples:
For m < n, let a = n2 − m2, b = 2mn, and c = n2 + m2; then a2 + b2 = c2.
Binomials that are sums or differences of cubes can be factored into smaller-degree polynomials as follows:
x
3
+
y
3
=
(
x
+
y
)
(
x
2
−
x
y
+
y
2
)
{\displaystyle x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})}
x
3
−
y
3
=
(
x
−
y
)
(
x
2
+
x
y
+
y
2
)
{\displaystyle x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})}
See also
Completing the square
Binomial distribution
List of factorial and binomial topics (which contains a large number of related links)
Notes
References
Bostock, L.; Chandler, S. (1978). Pure Mathematics 1. Oxford University Press. p. 36. ISBN 0-85950-092-6.
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Binomial (polynomial) - Wikipedia
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. [1] It is the simplest kind of a sparse polynomial after the monomials.
Monomials, Binomials, Trinomials and Polynomials - BYJU'S
What is the difference between monomials, binomials, trinomials and polynomials? A monomial has a single term, a binomial has two terms, a trinomial has three terms and a polynomial has one or more than one term.
Polynomials - Math is Fun
Monomial, Binomial, Trinomial. There are special names for polynomials with 1, 2 or 3 terms: How do you remember the names? Think cycles! There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used. Variables. Polynomials can have no variable at all
Binomial - Definition, Operations on Binomials & Examples - BYJU'S
In Mathematics, binomial is a polynomial that has two terms. An example of a binomial is x + 2. Visit BYJU'S to learn more about operations on binomials with solved examples.
Polynomials - Definition, Standard Form, Terms, Degree, Rules, …
Dec 19, 2024 · Based on the number of terms present in the expression, a polynomial is of 3 types: A polynomial consisting of only one term is called a monomial expression. Examples: 3x, 4xyz, and 2 x 2. A polynomial with two terms is known as a binomial expression. Examples: 2x + y, 4z + 7, and 10x 2 + 5x 3.
Types of Polynomials: Monomial, Binomial, Trinomial, Examples
Based on it, there are four major types of polynomials, zero or constant, linear, quadratic and cubic polynomial. Zero Polynomial. It does not have a variable. It has no terms. The coefficient of all variables is zero. Its degree is not defined. Zero polynomial is of the form P (x) = 0. Important Points to Remember:
Binomial Theorem - Math is Fun
We can use the Binomial Theorem to calculate e (Euler's number). e = 2.718281828459045... (the digits go on forever without repeating) It can be calculated using: (1 + 1/n) n (It gets more accurate the higher the value of n) That formula is a binomial, right? So let's use the Binomial Theorem: First, we can drop 1 n-k as it is always equal to 1:
Polynomial vs. Binomial — What’s the Difference?
Mar 28, 2024 · A binomial is a type of polynomial with exactly two terms, making it a subset of polynomials with a specific structure.
Binomial - Meaning, Coefficient, Factoring, Examples - Cuemath
In algebra, a binomial is an expression that has two unlike terms connected through an addition or subtraction operator in between. For example, 2xy + 7y is a binomial since there are two terms.
Binomial - Math.net
Multiplying binomials involves the use of the distributive property, as well as combining like terms to simplify the resulting expression. The distributive property states: a (b + c) = (a × b) + (a × c)