Binomial

Binomial Definition

The algebraic expression which contains only two terms is called binomial. It can also be stated that a  binomial is a two term polynomial. When a polynomial  is expressed as a sum or difference between two or more monomials it is said to be a  binomial. It is the simplest form of polynomial.

When expressed as a single indeterminate, a binomial can be expressed as \(ax^{m}+bx^{n}\)

Where a and b are numbers, and m and n are non negative distinct integers. x takes the form of a symbol known as indeterminate or a variable.

In Laurent polynomials, binomials are expressed similarly but the only difference is m and n can be negative. So the binomial from the previous example becomes

\(ax^{-m}+bx^{-n}\)

Binomial Examples

Some of the examples of a binomial are:

\(4x^{2}+5y^{2}\)
\(xy^{2}+xy\)
\(0.75x+\Theta y^{2}\)

Operations on Binomials

There are few basic operations that can be carried out on binomials.

Factorization:

Binomials can be factored and expressed as a product of other binomials.
For example, \(x^{2}-y^{2}\) can be expressed as \((x+y)(x-y)\).

Addition:

Addition of two binomials is done only when it contains like terms. This means that, it should have the same variable and same exponent.
For example,
Consider two binomials,
10x3 + 4y and 9x3 + 6y
Adding Binomials =(10x3 + 4y) + (9x3 + 6y)
Therefore, the resultant binomial = 19x3 + 10y

Subtraction:

Subtraction of two binomials is similar to the addition operation as if and only if it contains like terms. This means that, it should have the same variable and same exponent.
For example,
Consider two binomials,
12x3 + 4y and 9x3 + 10y
Subtracting Binomials =(12x3 + 4y) – (9x3 + 10y)
= 12x3 + 4y – 9x3 – 10y
Therefore, the resultant binomial = 3x3 – 6y

Multiplication:

When multiplying two binomials, distributive property is used and it ends up with four terms. It is generally referred as FOIL method, when the multiplication of two binomials are involved. Because, in this method multiplication of two binomials is carried out by multiplying each term of first factor/ binomial to the second factor/binomial. So, at the end multiplication of two binomial is expressed as a trinomial.

For example, the binomial \((mx+n)(ax+b)\) can be expressed as \(max^{2}+(mb+an)x+nb\)

Raising to nth Power

A binomial can be raised to nth power and you can find the detailed expansion of the general term \((x+y)^{n}\) under binomial expansion.

Converting to lower order binomials

Any higher order binomials can be factored down to lower order binomials such as cubes can be factored down to products of squares and another monomial.

For example, \(x^{3}+y^{3}\) can be expressed as \((x+y)(x^{2}-xy+y^{2})\)

Binomial Expansion

In Algebra, binomial theorem defines the algebraic expansion of the term \((x+y)^{n}\). It defines the power of each binomial in the form of \(ax^{b}y^{c}\). The exponents b and c are non-negative distinct integers and b+c = n and the coefficient a of each term is a positive integer and the value depends on n and b.

For example, for n=4, the expansion \((x+y)^{4}\) can be expressed as

\((x+y)^{4}\) = \(x^{4}\) +\(4x^{3}y\)+\(6x^{2}y^{2}\)+\(4xy^{3}\)+\(y^{4}\)

The coefficients of the binomials in this expansion 1,4,6,4, and 1 forms the 5th degree of Pascal’s triangle.

The general theorem for the expansion of \((x+y)^{n}\) is given as

\((x+y)^{n}\) = \({n \choose 0}x^{n}y^{0}\)+\({n \choose 1}x^{n-1}y^{1}\)+\({n \choose 2}x^{n-2}y^{2}\)+\(\cdots \)+\({n \choose n-1}x^{1}y^{n-1}\)+\({n \choose n}x^{0}y^{n}\)

Some of the methods used for the expansion of binomials are :

  • Pascal’s Triangle
  • Factorials
  • Combinations
  • Binomial Theorems
  • Binomial Series

Binomial Equation

Any equation that contains one or more binomial is known as a binomial equation.

Some of the examples of a binomial equation are:

\(x^{2}+2xy+y^{2}=0\) \(v=u+\frac{1}{2}at^{2}\)

Sample Example

Question:

Find the binomial from the following terms ?

  • 5x + 6y
  • 5 * y
  • 6xy
  • 6x ÷ y

Solution:

  • Option 1 : 5x + 6y : Here, addition operation makes the two terms from the polynomial
  • Option 2 : 5 * y : Multiplication operation produces 5y as a single term
  • Option 3 : 6xy : Multiplication operation produces the polynomial 6xy as a single term
  • Option 4 : 6x÷ y : Division operation makes the polynomial as single term
  • Therefore, the solution is 5x + 6y, is a binomial that has two terms.

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