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(a) 1, 5x - 4 2. 3x + 2 3. 10 + 2x(b) 1, 2x + 3y 2. 5x - y 3. x - 3y(c) 1, 2xy 2. xy 3. -7xy (d) 1. 2x3 -x2 + 6x - 8 2. 10 + 7x - 2x3 + 4x3
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If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Binomial is a polynomial with only terms. For example, x + 2 is a binomial, where x and 2 are two separate terms. Also, the coefficient of x is 1, the exponent of x is 1 and 2 is the constant here. Therefore, A binomial is a two-term algebraic expression that contains variable, coefficient, exponents and constant. Another example of a binomial polynomial is x2 + 4x. Thus, based on this binomial we can say the following:
Is 2x a binomial? 2x has only one term, hence it is not a binomial. Binomial DefinitionThe 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.
axm + bxn 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.
ax-m + bx-n Examples of BinomialSome of the binomial examples are;
Other PolynomialsApart from the binomial, the other two types of the polynomial are: Monomial: When an expression is having only one term or a single term, then such polynomial is known as a monomial. Examples of monomial are 3x, 4, 5x2, 6x3, etc. Trinomial: A trinomial is a polynomial that has only three terms. For example, x2 – 3 + 3x. Binomial EquationAny equation that contains one or more binomial is known as a binomial equation. Some of the examples of this equation are: x2 + 2xy + y2 = 0 v = u+ 1/2 at2 Operations on BinomialsThere are few basic operations that can be carried out on this two-term polynomials are:
Factorization of BinomialsWe can factorise and express a binomial as a product of the other two. Addition of BinomialsAddition 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. 10x3 + 4y and 9x3 + 6y Adding both the equation = (10x3 + 4y) + (9x3 + 6y) Therefore, the resultant equation = 19x3 + 10y Subtraction of BinomialsSubtraction of two binomials is similar to the addition operation as if and only if it contains like terms. For example, 12x3 + 4y and 9x3 + 10y Subtracting the above polynomials, we get; (12x3 + 4y) – (9x3 + 10y) Multiplication of BinomialsWhen 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 max2+(mb+an)x+nb Raising to nth PowerA binomial can be raised to the nth power and expressed in the form of; (x + y)n Converting to lower-order binomialsAny 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, x3 + y3 can be expressed as (x+y)(x2-xy+y2) Binomial ExpansionIn Algebra, binomial theorem defines the algebraic expansion of the term (x + y)n. It defines power in the form of axbyc. 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 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 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 = \(\begin{array}{l}{n \choose 0}x^{n}y^{0}\end{array} \) +\(\begin{array}{l}{n \choose 1}x^{n-1}y^{1}\end{array} \) +\(\begin{array}{l}{n \choose 2}x^{n-2}y^{2}\end{array} \) +\(\begin{array}{l}\cdots \end{array} \) \(\begin{array}{l}{n \choose n-1}x^{1}y^{n-1}\end{array} \) +\(\begin{array}{l}{n \choose n}x^{0}y^{n}\end{array} \) Some of the methods used for the expansion of binomials are :
Binomial FormulaBinomial DistributionThe term binomial distribution is used for a 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 SolutionsQuestion 1: Find the binomial from the following terms? Solution:
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 + 8x2 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 2x2 + 6x and 3x2 – 2x. Solution: 2x2 + 6x + 3x2 − 2x Arrange the like terms 2x2 + 3x2 + 6x − 2x (2+3)x2 + (6−2)x 5x2 + 4x Learn more about binomials and related topics in a simple way. Register with BYJU’S – The Learning App today. A binomial is a polynomial or an algebraic expression that contains only two terms. The examples of binomial are 3x + 2, 2×2 + x, x + y, etc. A binomial have only positive exponents and not negative exponents. x+5 is a binomial, because x is a variable and 5 is a constant. A term can be a variable, constant, or the combination of variable and constant. The degree of binomial is the largest exponent. Therefore, the degree of x3 + 3x2 is 3. |