Question

How do you expand ${(x + 3)^6}$ using Pascal’s Triangle?

Answer 1

Hint: According to the question we have to determine the expansion of ${(x + 3)^6}$ which is as mentioned in the question using Pascal’s Triangle. So, first of all to determine the expansion of using Pascal’s Triangle we have to understand about the Pascal Triangle which is as explained below:
Pascal Triangle: It is a kind of number pattern. The numbers are so arranged that they reflect a triangle. First number 1 is placed on the top and then we start putting the numbers in a triangular form. The numbers which we can get in each step are the addition of the above two numbers. It is similar to the concept of triangular numbers.
Now, we to expand the given polynomial we have to use the formula to find the expansion using Pascal triangle which is as mentioned below:

Formula used: $ {(x + y)^6} = {x^6} + 6{x^5}y + 15{x^4}{y^2} + 20{x^3}{y^3} + 15{x^2}{y^4} + 6x{y^5} + {y^6}.................(A)$
Now, we have to substitute the value of y as 3 in the formula (A) which is as mentioned above.
Now, we have to add or subtract the term which can be added or subtracted to simplify the expansion obtained.

Complete step-by-step solution:
Step 1: First of all to determine the expansion of using Pascal’s Triangle we have to understand about the Pascal Triangle which is as explained in the solution hint.
Step 2: Now, to expand the given polynomial we have to use the formula to find the expansion using the Pascal triangle which is as mentioned in the solution hint. Hence,
$ \Rightarrow {(x + y)^6} = {x^6} + 6{x^5}y + 15{x^4}{y^2} + 20{x^3}{y^3} + 15{x^2}{y^4} + 6x{y^5} + {y^6}$
Step 3: Now, we have to substitute the value of y as 3 in the formula (A) which is as mentioned in the solution hint. Hence,
$ \Rightarrow {(x + 3)^6} = {x^6} + 6{x^5}3 + 15{x^4}{3^2} + 20{x^3}{3^3} + 15{x^2}{3^4} + 6x{3^5} + {3^6}$
Step 4: Now, we have to add or subtract the term which can be added or subtracted to simplify the expansion obtained. Hence,
\[ \Rightarrow {(x + 3)^6} = {x^6} + 18{x^5} + 135{x^4} + 540{x^3} + 1215{x^2} + 1458x + 729\]

Hence, with the help of the formula (A) we have determined the expansion using ${(x + 3)^6}$ Pascal triangle is \[ \Rightarrow {(x + 3)^6} = {x^6} + 18{x^5} + 135{x^4} + 540{x^3} + 1215{x^2} + 1458x + 729\].

Note: According to Pascal, the triangle is a kind of number pattern. The numbers are so arranged that they reflect a triangle. First number 1 is placed on the top and then we start putting the numbers in a triangular form. The numbers which we can get in each step are the addition of the above two numbers.
On substituting the value of 3 in place of y in the formula (A) we can easily determine the required expansion using the Pascal triangle.