A binomial is a mathematical expression generally found in algebra that includes two unlike terms connected by the operators plus or minus. The terms consist of variables and constants. For example, 3x + 8 is a binomial which has two terms 3x and 8.Binomial theorem describes the process of deriving the expanded form of the algebraic expression involving two variables x and y which can be in the form of (x + y) raised to the power of n. Here n is called the exponent which indicates how many times an expression is multiplied by itself. The exponent value of this binomial theorem expansion can be any integer including a negative number or a fraction. The value of an expression with a smaller value of exponent can be obtained by simple algebraic multiplication multiple times as denoted by the exponent value. But for an algebraic expression with a higher exponent value, a rule has to be followed to find the expanded form. This rule is known as the binomial theorem.
Explanation
The binomial theorem states the principle or process for the expansion of a binomial expression with an exponent. As per the binomial theorem, a binomial in the form (x + y) having a non-negative exponent can be expanded into a sum of different terms involving variables x and y with each variable having an individual exponent’s value. The terms in a binomial expanded form also can have a numeric value attached to the variables. This value is called a coefficient. For example, in the binomial expression 4x+5, the coefficient of x is 4.
Properties of the Binomial Theorem
- The expanded form of a binomial expression contains one term more than the number given as the power or exponent of the binomial. For example, if the power of the binomial expression is 3, then the expanded form will contain 4 terms.
- The sum of exponents of each term in the expanded form is equal to the power of the binomial.
Importance of Binomial Theorem
The value of an expression with a smaller value of exponent can be obtained easily as it involves simple algebraic calculation by multiplying repeatedly depending on the given exponent value. But for an algebraic expression with a higher exponent value, there has to be a rule that states how to derive the expanded form and get the value of the expression. This rule is known as the binomial theorem which has wide application in various fields of mathematics as well as science.
Points of Binomial Theorem
- The number of terms in the expanded form of a binomial expression will be one more than the numeric value given as the exponent of the binomial. For example, if the power of the binomial expression is 3, then the expanded form of this expression will contain 4 terms.
- The sum of exponents of each term in the expanded form of binomial expression is equal to the power of the given binomial expression.
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