PASCAL TRIANGLE ALGEBRAIC EXPANSION

About "Pascal Triangle Algebraic Expansion"

Pascal Triangle Algebraic 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. 

Working Rule 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 4th 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)

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 + 6a2b24ab3 + b4

Working Rule 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)

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 + 6a2b24ab3 + b4

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

Some More Examples

Example 1 :

Expand the following using pascal triangle

(3x + 4y)4

Solution : 

Already, we know  

(a + b)4  =  a4 + 4a3b + 6a2b24ab3 + b4

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

a  =  3x,  b  =  4y

Let us plug 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)24(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

Example 2 :

Expand the following using pascal triangle

(x - 4y)4

Solution : 

Already, we know

(a - b)4  =  a4 - 4a3b + 6a2b24ab3 + b4

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

a  =  x,  b  =  4y

Let us plug a  =  x,  b  =  4y   in the expansion of (a - b)

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

After having gone through the stuff and examples explained, we hope that the students would have understood,  "Pascal triangle algebraic expansion".

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