Binomial Theorem

After completing this tutorial you will have a quick and easy method of expanding a binomial of any degree with the form (a + b)ⁿ 

Polynomial Multiplication Patterns

1.      (a + b)²  =  a² + 2ab + b²

2.      (a – b)²  =   a² - 2ab + b²

3.      (a + b)(a – b) =  a² – b²

4.      (a + b)³ =  a³ + 3a² b + 3ab² + b³

5.      (a - b)³ =  a³ - 3a² b + 3ab² + b³

Binomial Expansion Pattern

     It is possible to write the expansion of (a + b)ⁿ, where n is any positive integer, without showing all of the steps of multiplying and combining similar terms.

(a + b)¹ =  a + b

(a + b)²  =  a² + 2ab + b²

(a + b)³ =  a³ + 3a² b + 3ab² + b³

(a + b)⁴ =  a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

(a + b)⁵ =  a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

(a + b)⁶ =  a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶

      The exponents of “a” begin with the exponent of the binomial and decrease by 1, term by term. There is always one more term than the degree of the binomial.

      The exponents of “b” begin with 0 and increase by 1, term by term. Until the last term until the last term which contains “b” is the degree of the binomial.

      The variables in the expansion of (a + b)ⁿ have the pattern;-

aⁿ, a^n-1b, a^n-2b², . . . , ab^{n -1}, bⁿ

When we arrange the coefficients in triangular formation it yields an easy to remember pattern.

                              

This triangular array is called Pascal’s Triangle.

-          The exponent of “a” begins with the degree of the binomial and decreases to 0

-          The exponent of “b” begins with 0 and increases to the degree of the binomial

Using Pascal’s Triangle for the coefficients expand the following;-

(a – b)⁴

This can be rewritten as an addition with “b” as a negative number, and this will help us see it as an addition. Using Pascal’s Triangle and what we have just learned about the binomial expansion pattern we can write out the exponents.

[a + (-b)]⁴ = a⁴ + 4a³(-b) + 6a²(-b)² + 4a(-b)³ + (-b)⁴

            We used Pascal’s Triangle on the fourth row to get the coefficients, we just finish up the math to get which terms are negative and which are positive. The negative terms will all have odd exponents on the “b” term.

a⁴ + 4a³(-b) + 6a²(-b)² + 4a(-b)³ + (-b)⁴ = a⁴ - 4a³b + 6a²b² - 4ab³ + b⁴

Binomial Theorem

We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As you may have seen, multiplication can be time-consuming or even not possible in some cases.

Properties of the Binomial Expansion (a + b)ⁿ

• There are n + 1 terms. That is to say, there is one more term than the degree of the binomial
• The first term is aⁿ and the final term is bⁿ.
• Progressing from the first term to the last, the exponent of a decreases by 1 from term to term while the exponent of b increases by 1. In addition, the sum of the exponents of a and b in each term is n. Tha6t is to say, that the sum of the exponents of “a” and “b” is equal to the exponent of the binomial.
• If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.

General formula for (a + b)ⁿ

First, we need the following definition:                                                                    

Definition: n! represents the product of the first n positive integers i.e.                  

 n! = n(n − 1)(n − 2) ... (3)(2)(1)                                                                                    

We say n! as 'n factorial'

Here are some factorial values:

 3! = (3)(2)(1) = 6                                                                                                                   

 5! = (5)(4)(3)(2)(1) = 120

                                                Note :        cannot be canceled down to2!

Binomial Theorem Formula

Based on the binomial properties, the binomial theorem states that the following binomial formula is valid for all positive integer values of n:

This can be written more simply as:

(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^{n − 1}b + ⁿC₂a^{n − 2}b² + ⁿC₃a^{n − 3}b³ + ... + ⁿC_nbⁿ

                                               We can use the               button on our calculator to find these values.

Using the binomial theorem, expand (x + 2)⁶.

(x + 2)⁶ = x⁶ + 6x⁵2¹ + 15x⁴2² + 20x³2³ + 15x²2⁴ + 6x¹2⁵ + 2⁶

               ↓ 

             (1 x 6) = 6 
                 1
  Here I took the coefficient of the first term 1 and multiplied it by the

exponent of the “a” term 6, then divided it by the number of the term 1, to get the coefficient of the next term. I do the same to the next term  
(6 x 5) = 15

    2

Therefore 15 is the coefficient of the third term. I continue this process until I reach the last term bⁿ.

Finishing up the math we get;-

= x⁶ + 12x⁵ + 60x⁴ + 160x³ + 240x² + 192x + 64