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Question: How do you find the coefficient of \({x^2}\)in the expansion of\({(x + 2)^5}\)?...

How do you find the coefficient of x2{x^2}in the expansion of(x+2)5{(x + 2)^5}?

Explanation

Solution

According to the question we have to determine the coefficient of x2{x^2}in the expansion of(x+2)5{(x + 2)^5}. So, first of all to determine the coefficient ofx2{x^2}in the given expansion we have to use the formula to determine the expansion having whole power 5 which is as mentioned below:

Formula used:
(x+y)5=x5+5x4y+10x3y2+10x2y3+5xy4+y5.................(A){(x + y)^5} = {x^5} + 5{x^4}y + 10{x^3}{y^2} + 10{x^2}{y^3} + 5x{y^4} + {y^5}.................(A)
Hence, with the help of the formula above we can easily determine the required coefficient of the expansion which is (x+2)5{(x + 2)^5}
Now, after finding the expansion of (x+2)5{(x + 2)^5}we have to solve the expression obtained means we have to multiply or divide the terms of the expansion.
Now after multiplication or division we have to obtain the required coefficient of x2{x^2}.

Complete step by step solution:
Step 1: First of all to determine the coefficient ofx2{x^2}in the given expansion we have to use the formula to determine the expansion having whole power 5 which is as mentioned in the solution hint. Hence,
⇒(x−2)5=x5+5×2x4+10x3(2)2+10x2(2)3+5x(2)4+(2)5\Rightarrow {(x - 2)^5} = {x^5} + 5 \times 2{x^4} + 10{x^3}{(2)^2} + 10{x^2}{(2)^3} + 5x{(2)^4} + {(2)^5}
Step 2: Now, after finding the expansion of (x+2)5{(x + 2)^5}we have to solve the expression obtained means we have to multiply or divide the terms of the expansion. Hence,

⇒(x−2)5=x5+10x4+10x3(4)+10x2(8)+5x(16)+(32) ⇒(x−2)5=x5+10x4+40x3+80x2+80x+(32)  \Rightarrow {(x - 2)^5} = {x^5} + 10{x^4} + 10{x^3}(4) + 10{x^2}(8) + 5x(16) + (32) \\\ \Rightarrow {(x - 2)^5} = {x^5} + 10{x^4} + 40{x^3} + 80{x^2} + 80x + (32) \\\

Step 3: Now after multiplication or division we have to obtain the required coefficient ofx2{x^2}. Hence, required coefficient is,
=80= 80

Hence, with the help of the formula (A) which is as mentioned in the solution hint we have determined the required coefficient of x2{x^2}in the expansion of(x+2)5{(x + 2)^5}which is=80 = 80.

Note:
To determine the coefficient of x2{x^2} it is necessary that we have to expand the given expression with the help of the formula of expansion having whole power which is 5 with the help of the formula (A) as mentioned in the solution hint.
A coefficient is a number or an integer which is a product of that variable of which it is coefficient.