**Pascal Triangle and Binomial Expansion Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on expanding binomials using pascal triangle.

Before look at the worksheet, if you would like learn the stuff related to Pascal's Triangle and the Binomial Theorem,

**Problem 1 :**

Expand the following using pascal triangle

(3x + 4y)^{4}

**Problem 2 :**

Expand the following using pascal triangle

(x - 4y)^{4}

**Problem 1 :**

Expand the following using pascal triangle

(3x + 4y)^{4}

**Solution : **

Pascal's Triangle :

In (3x + 4y)^{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

**Step 1 :**

In the first term, we have to take only "3x" with power "4" [This is the exponent of (3x + 4y)].

Then, the first term is

= (3x)^{4}

= 81x^{4}

**Step 2 :**

In the second term, we have to take both "3x" and "4y".

For "3x", we have to take exponent "1" less than the exponent of "3x" in the previous term.

For "4y", we have to take exponent "1".

Then, the second term is

= (3x)^{3}(4y)

= (27x^{3})(4y)

= 108x^{3}y

**Step 3 : **

In the third term also, we have to take both "3x" and "4y".

For "3x", we have to take exponent "1" less than the exponent of "3x" in the previous term.

For "4y", we have to take exponent "2".

Then, the second term is

= (3x)^{2}(4y)^{2}

= (9x^{2})(16y^{2})

= 144x^{2}y^{2}

(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 "3x" will be the last term.

In the last term, we will have only "4y" with power "4" [This is the exponent of (3x + 4y)]

Then, the last term is

= (4y)^{4}

= 256y^{4}

The four steps explained above have been summarized in the diagram shown below.

Hence, the expansion of (3x + 4y)^{4 }is

(3x + 4y)^{4 }= 81x^{4} + 432x^{3}y + 864x^{2}y^{2} + 768xy^{3} + 256y^{4}

**Problem 2 :**

Expand the following using pascal triangle

(x - 4y)^{4}

**Solution :**

We can follow the steps explained in problem 1 and get the expansion of (x - 4y)^{4.}

But we have negative sign in (x - 4y)^{4}.

So, we have to take positive and negative signs alternatively staring with positive sign for the first term.

Hence, the expansion of (x - 4y)^{4}^{ }is

(x - 4y)^{4} = x^{4} - 16x^{3}y + 96x^{2}y^{2} - 256xy^{3} + 256y^{4}

After having gone through the stuff and examples explained, we hope that the students would have understood how to expand a binomial using Pascal triangle.

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