Binomial is a polynomial with only terms. For example, x + 2 is a binomial, where x and 2 are two separate terms. A binomial is an algebraic expression, that contains variable, coefficient, exponents and constant.
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Binomial Definition
The algebraic expression which contains only two terms is called binomial. It is a two-term polynomial. Also, it is called a sum or difference between two or more monomials. It is the simplest form of a polynomial.
- When expressed as a single indeterminate, a binomial can be expressed as;
ax^{m} + bx^{n}
Where a and b are the numbers, and m and n are non-negative distinct integers. x takes the form of indeterminate or a variable.
- In Laurent polynomials, binomials are expressed in the same manner, but the only difference is the exponents, m and n can be negative. Therefore, we can write it as;
ax^{-m} + bx^{-n}
Examples of Binomial
Some of the binomial examples are;
- 4x^{2}+5y^{2}
- xy^{2}+xy
- 0.75x+10y^{2}
- x+y
- x^{2} + 3
- x^{3} + 2x
- 9 + 7y
- m + 2n
Other Polynomials
Apart from the binomial, the other two types of the polynomial are:
- Monomial
- Trinomial
When an expression is having only one term or single term, then such polynomial is known as a monomial. Examples of monomial are 3x, 4, 5x^{2}, 6x^{3}, etc.
A trinomial is a polynomial that has only three terms. For example, x^{2} – 3 + 3x.
Binomial Equation
Any equation that contains one or more binomial is known as a binomial equation.
Some of the examples of this equation are:
x^{2 }+ 2xy + y^{2 }= 0
v = u+ 1/2 at^{2}
Operations on Binomials
There are few basic operations that can be carried out on this two-term polynomials are:
- Factorisation
- Addition
- Subtraction
- Multiplication
- Raising to n^{th} Power
- Converting to lower-order binomials
Factorization
We can factorise and express a binomial as a product of the other two.
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 the same exponent.
For example,
Let us consider, two equations.
10x^{3} + 4y and 9x^{3} + 6y
Adding both the equation = (10x^{3} + 4y) + (9x^{3} + 6y)
Therefore, the resultant equation = 19x^{3} + 10y
Subtraction
Subtraction of two binomials is similar to the addition operation as if and only if it contains like terms.
For example,
12x^{3} + 4y and 9x^{3} + 10y
Subtracting the above polynomials, we get;
(12x^{3} + 4y) – (9x^{3} + 10y)
= 12x^{3} + 4y – 9x^{3} – 10y
Therefore, the resultant equation is = 3x^{3} – 6y
Multiplication
When multiplying two binomials, the distributive property is used and it ends up with four terms. It is generally referred to as the FOIL method. Because in this method multiplication is carried out by multiplying each term of the first factor to the second factor. So, in the end, multiplication of two two-term polynomials is expressed as a trinomial.
For example, (mx+n)(ax+b) can be expressed as max^{2}+(mb+an)x+nb
Raising to nth Power
A binomial can be raised to the nth power and expressed in the form of;
(x + y)^{n}
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 power 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 Formula
\((a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) a^{n-k} b^{k}\) |
Binomial Distribution
The term binomial distribution is used for discrete probability distribution. There are only two outcomes here Success and Failure. Learn in detail Binomial distribution and binomial distribution formula at BYJU’S.
Problems and Solutions
Question 1: 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 a single term.
Therefore, the solution is 5x + 6y, is a binomial that has two terms.
Question 2: Multiply (5 + 4x) . (3 + 2x).
Solution: (5 + 4x)(3 + 2x)
= (5)(3) + (5)(2x) + (4x)(3) + (4x)(2x)
= 15 + 10x + 12x + 8(x)^{2}
= 15 + 22x + 8x^{2}
Question 3: Add: 6a + 8b – 7c, 2b + c – 4a and a – 3b – 2c.
Solution: (6a + 8b – 7c) + (2b + c – 4a) + (a – 3b – 2c)
= 6a + 8b – 7c + 2b + c – 4a + a – 3b – 2c
Arranging the like terms, we get;
= 6a – 4a + a + 8b + 2b – 3b – 7c + c – 2c
= 3a + 7b – 8c
Question 4:Add 2x^{2} + 6x and 3x^{2} – 2x.
Solution: 2x^{2} + 6x + 3x^{2} − 2x
Arrange the like terms
2x^{2 }+ 3x^{2} + 6x − 2x
(2+3)x^{2} + (6−2)x
5x^{2} + 4x
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