PASCAL'S TRIANGLE AND BINOMIAL EXPANSION

This is one of the good applications in mathematics that can be used to expand binomial expressions with any power. 

Let us see how pascal triangle can be used to expand

(a + b)ⁿ

for any value of "n"

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, b = 4y in the expansion of (a + b)⁴.

So, we have

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

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

(3x + 4y)⁴ = 81x⁴ + 432x³y + 864x²y² + 768xy³ + 256y⁴

Problem 2 :

Expand the following using pascal triangle

(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.

So, we have

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

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

(x - 4y)⁴ = x⁴ - 16x³y + 96x²y² - 256xy³ + 256y⁴

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