PASCAL'S TRIANGLE AND THE BINOMIAL THEOREM 

A binomial expression is the sum, or difference, of two terms.

For example,

x + 2,  2x + 3y,  p - q

We may already be familiar with the need to expand brackets when squaring such quantities.

We will know, for example, that

(x + 3)² = (x + 3)(x + 3)

(x + 3)² = x² + 3x + 3x + 9

(x + 3)² = x² + 6x + 9

If we want to raise a binomial expression to a power higher than 2. 

For example if we want to find (x + 3)⁷, it is bit difficult to do this by repeatedly multiplying (x + 3) by itself. 

In this section, we will learn how a triangular pattern of numbers, known as Pascal’s triangle, can be used to obtain the required result very quickly.

Pascal's Triangle

We can form a Pascal's triangle using the steps explained below.

Step 1 :

We start to generate Pascal’s triangle by writing down the number 1.

Then we write a new row with the number 1 twice :

Step 2 :

We then generate new rows to build a triangle of numbers. Each new row must begin and end with a 1 :

Step 3 :

The remaining numbers in each row are calculated by adding together the two numbers in the row above which lie above-left and above-right.

So, adding the two 1’s in the second row gives 2, and this number goes in the vacant space in the third row :

Step 4 :

The two vacant spaces in the fourth row are each found by adding together the two numbers in

the third row which lie above-left and above-right :

1 + 2  =  3 and 2 + 1  =  3

This gives :

Step 5 :

We can continue to build up the triangle in this way to write down as many rows as we wish.

The diagram below shows the first six rows of Pascal’s triangle.

Pascal Triangle and Exponent of the Binomial

To understand pascal triangle algebraic expansion, let us consider the expansion of (a + b)⁴ using the pascal triangle given above. 

How to Get Expansion of (a + b)⁴ Using Pascal Triangle

In (a + b)⁴, the exponent is '4'. 

So, let us take the row in the above pascal triangle which is corresponding to 4^th power. 

That is,  

1    4    6    4    1

General Rule :

In pascal expansion, we must have only 'a' in the first term, only 'b' in the last term and 'ab' in all other middle terms. 

If we are trying to get expansion of (a + b)ⁿ, all the terms in the expansion will be positive. 

Note :

This rule is not only applicable for power '4'.

This rule is applicable for any value of 'n'  in (a + b)ⁿ. 

It has been clearly explained below.

Now we have to follow the steps given below. 

Step 1 :

In the first term, we have to take only 'a' with power '4' [This is the exponent of (a + b)].

Then, the first term will be a⁴.

Step 2 :

In the second term, we have to take both 'a' and 'b'. 

For 'a', we have to take exponent '1' less than the exponent of 'a' in the previous term. 

For 'b', we have to take exponent '1'.

Then, the second term will be  a³b.

Step 3 : 

In the third term also, we have to take both 'a' and 'b'. 

For 'a', we have to take exponent '1' less than the exponent of 'a' in the previous term. 

For 'b', we have to take exponent '2'.

Then, the second term will be a²b².

(We have to continue this process, until we get the exponent '0' for 'a')

Step 4 : 

When we continue the process said in step 3, the term in which we get exponent '0' for 'a' will be the last term.

In the last term, we will have only 'b' with power '4' [This is the exponent of (a + b)].

Then, the last term will be b⁴.

The four steps explained above given in the picture below. 

Finally the expansion is,

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

How to Get Expansion of (a - b)⁴ Using Pascal Triangle

General Rule :

If we are trying to get expansion of (a - b)ⁿ, we have to take positive and negative signs alternatively staring with positive sign for the first term. 

Note :

This rule is not only applicable for power '4'. This rule is applicable for any value of 'n' in (a - b)ⁿ. 

To get expansion of (a - b)⁴, we do not have to do much work.

As we have explained above, we can get the expansion of (a + b)⁴ and then  we have to take positive and negative signs alternatively staring with positive sign for the first term

So, the expansion is 

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

In this way, using pascal triangle to get expansion of a binomial with any exponent. 

Solved Problems

Expand the following binomials using pascal triangle :

Problem 1 :

(3x + 4y)⁴

Solution :

Already, we know

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

Comparing (3x + 4y)⁴ and (a + b)⁴, we get 

a = 3x  and  b = 4y

Substitute a = 3x and b = 4y in the expansion of (a + b)⁴

(3x + 4y)⁴

= (3x)⁴ + 4(3x)³(4y) + 6(3x)²(4y)² + 4(3x)(4y)³ + (4y)⁴

= 81x⁴ + 4(27x³)(4y) + 6(9x²)(16y²) + 4(3x)(64y³) + 256y⁴

= 81x⁴ + 432x³y + 864x²y² + 768xy³ + 256y⁴

Problem 2 :

(x - 4y)⁴

Solution :

Already, we know

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

Comparing (x - 4y)⁴ and (a - b)⁴, we get 

a = x  and  b = 4y

Substitute a = x and b = 4y in the expansion of (a - b)⁴.

(x - 4y)⁴ = x⁴ - 4(x³)(4y) + 6(x²)(4y)² - 4(x)(4y)³ + (4y)⁴

= x⁴ - 16x³y + 6(x²)(16y²) - 4(x)(64y³) + 256y⁴

= x⁴ - 16x³y + 96x²y² - 256xy³ + 256y⁴

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