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. Algebraic expressions can be categorized into different types depending upon the number of terms present, like monomial, binomial, trinomial, etc.
In this article, we will explore the binomial expression in algebra, its properties and its identities that are used to solve various problems in algebra. We shall go through different solved examples based on binomial for a better understanding of the concept.
| 1. | What is a Binomial? |
| 2. | Binomial Meaning |
| 3. | Binomial Coefficient |
| 4. | Factoring Binomial |
| 5. | Squaring Binomial |
| 6. | FAQs on Binomial |
A binomial is an algebraic expression that has two terms. In other words, an algebraic expression consisting of two unlike terms having constants and variables is a binomial expression. These terms are joined using arithmetic operators such as + (plus) and –(minus). A binomial, along with monomial, trinomial, quadrinomial, etc is categorized under algebraic expressions based on the number of terms it contains. Observe the following unlike terms mentioned in the image below. These are a few illustrations explaining what and how exactly polynomials are categorized as binomial, monomial, and trinomial.
Binomial is an algebraic expression that contains two different terms connected by addition or subtraction. In other words, we can say that two distinct monomials of different degrees connected by plus or minus signs form a binomial. For example, consider two monomials, 2x and 5x 10 . The expression to add these monomials gives a binomial given by, 2x + 5x 10 .
Binomial coefficients are the positive integers that are the coefficients of terms in a binomial expansion. We know that a binomial expansion '(x + y) raised to n' or (x + n) n can be expanded as, (x+y) n = n C0 x n y 0 + n C1 x n-1 y 1 + n C2 x n-2 y 2 + . + n Cn-1 x 1 y n-1 + n Cn x 0 y n , where, n ≥ 0 is an integer and each n Ck is a positive integer known as a binomial coefficient using the binomial theorem. Here, n is the power of the binomial, and k is 1 less than the number of the term we are considering where n ≥ k ≥ 0. The formula to find the binomial coefficient of the k th term of any binomial raised to power n is given below,
Let us take an example to understand it better. To find the binomial coefficients of the expansion (x + 4) 5 , let us apply the above binomial coefficient formula. Here, the value of n is 5.
For k = 0 (for the first term), we have, 5 C0 = (5!) / [0! (5-0)!]
It implies that the binomial coefficient of the first term of the expression (x + 4) 5 is 1. Similarly, for k=1 (for the second term), we have, 5 C1 = (5!) / [1! (5-1)!].
It means the binomial coefficient of the second term of the expression (x + 4) 5 is 5. Similarly,
⇒ 5 C2 = 5!/(2! × 3!) = (5 × 4 × 3!)/(2 × 1 × 3!)
⇒ 5 C3 = 5!/(3! × 2!) = (5 × 4 × 3!)/(3! × 2 × 1)
Using this binomial coefficients formula, we can find out the coefficients of the terms without even expanding the expansion. The coefficients of all the 6 terms of the binomial (x + 4) 5 are 1, 5, 10, 10, 5, and 1. One interesting fact here is that if we find and arrange the binomial coefficients of the expansion in the triangle form, we will get a special type of triangle known as Pascal's triangle.
Factoring is the process of expressing an algebraic expression as a product of its factors. Factoring binomial means breaking down the binomial into the product of two expressions. As we know that binomials are expressions containing two terms, so by factoring a binomial, we will get its two factors of a lower degree. There are four rules of factoring binomial which are given below:
If both the terms of the given binomial have a common factor, then it can be used to factor the binomial. For example, in 2x 2 + 6x, both the terms have a greatest common factor of 2x.
When 2x 2 ÷ 2x = x and, 6x ÷ 2x = 3
Therefore, 2x 2 + 6x can be factored as 2x(x + 3).
With some binomials, there are no common factors of both the terms, but still, we can factorize them. One such way is by considering the difference of squares. If we recognize that both the terms are in the form of x 2 - y 2 , then, we can use the following identity to factorize such binomials: x 2 - y 2 = (x+y)(x-y). For example, let us factorize a 2 - 9. Here, a 2 is the square of a, and 9 is the square of 3.
⇒ a 2 - 9 = a 2 - 3 2
By using the algebraic identity: x 2 - y 2 = (x+y)(x-y), we can write it as (a+3)(a-3). Therefore, a 2 - 9 = (a+3)(a-3).
Sometimes, binomials are given as the sum of cubes, for example, x 3 + 27. In such cases, the following algebraic identity can be used to factorize the binomial: a 3 + b 3 = (a + b)(a 2 - ab + b 2 ). For example, let us factorize the binomial x 3 + 27. Here x 3 is the cube of x and 27 is the cube of 3.
⇒ x 3 + 27 = x 3 + 3 3
By using the algebraic identity: a 3 + b 3 = (a + b)(a 2 - ab + b 2 ), we can write it as (x + 3)(x 2 - 3x + 9). Therefore, x 3 + 27 = (x + 3)(x 2 - 3x + 9).
Another type of binomial is the difference of cubes, for example, y 3 - 64. In such cases, the following algebraic identity can be used to factorize the binomial: a 3 - b 3 = (a - b)(a 2 + ab + b 2 ). For example, let us factorize the binomial y 3 - 64. Here y 3 is the cube of y and 64 is the cube of 4.
⇒ y 3 - 64 = y 3 - 4 3
By using the algebraic identity: a 3 - b 3 = (a - b)(a 2 + ab + b 2 ), we can write it as (y - 4)(y 2 + 4y + 16). Therefore, y 3 - 64 = (y - 4)(y 2 + 4y + 16).
Here, it is important to note that every binomial cannot be factored into two expressions containing rational coefficients. It should be either of the four types explained above. One example of a binomial that cannot be factored is 3a 2 + 16.
The square of a binomial is the sum of the square of the first term, twice the product of both terms, and the square of the second term. When the sign of both terms is positive, then we use the following identity for squaring binomial: (a + b) 2 = a 2 + 2ab + b 2 . When the sign of the second term is negative, then we use the following identity: (a - b) 2 = a 2 - 2ab + b 2 . And, when both the terms are negative, then the following identity is to be used for squaring binomial: (- a - b) 2 = a 2 + 2ab + b 2 . Let us take an example to understand the concept of squaring binomial. Find the square of 2x - 5.
⇒ (2x - 5) 2 = (2x) 2 - 2 × (2x) × 5 + (5) 2 [By using the identity: (a - b) 2 = a 2 - 2ab + b 2 ]
Therefore, (2x - 5) 2 = 4x 2 - 20x + 25. Here, it is important to note that the square of a binomial is always a trinomial.
Important Notes on Binomial
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Example 1: Choose the binomials from the following expressions: (a) x 2 (b) 3 + 5x (c) x+5y. Solution: The expressions (b) 3 + 5x and (c) x+5y are binomials as these expressions have exactly two terms.
Example 2: Find the binomial coefficient of the 5th term of the expansion of (a - 9) 8 . Solution: The formula to find the binomial coefficient is n Ck = (n!) / [k ! (n-k)!]. Here, n is the exponent of the given expression and k is 1 less than the term we are considering. In the given question, n = 8, and k = 5 - 1 = 4. 8 C4 = (8!) / [4 ! (8-4)!] = 8!/(4! × 4!) = (8 × 7 × 6 × 5 × 4!)/(4 × 3 × 2 × 1 × 4!) = (8 × 7 × 6 × 5)/(4 × 3 × 2 × 1) = 70
Thus, the coefficient of the 5th term of the expansion of (a - 9) 8 is 70.
Example 3: Simplify the following sum of binomial expressions: (7x + 9y) + (-9 + 2x). Solution: (7x + 9y) + (-9 + 2x)
Separating the like terms,
= 7x + 2x + 9y - 9
= 9x + 9y - 9
Taking 9 as a common factor,
9(x + y -1)
Thus, simplifying the binomial expressions (7x + 9y) + (-9 + 2x) gives 9(x + y -1) as the result.
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