This is one of the famous arguments in algebra. Remember the argument why doesn’t (x+y)2 equal x2+y2? (x+y)2 (x+y)2 and x2+y2x2+y2 are both algebraic expression. An algebraic expression is made up of variable, integer constants and an arithmetic operator. In this example, x and y are variables, + is an arithmetic operator.
There is another arithmetic operator in the expression (x+y)2 (x+y)2, which is the multiplication operator. In algebra multiplication operator are not denoted with “*”. In other words, the expression xy is equal to x*y. Similarly (x)(y) is also equal to xy. It is tempting to say that (x+y)2 (x+y)2 does equal x2+y2x2+y2 by the looks of the equation. However, the law of algebra proves that this hypothesis incorrect. You need to be familiar with algebraic multiplication rule in order to solve this. It is similar to that of Distributive Law. In other words, multiply each variable inside the bracket with the variable outside the bracket. Let us now see how (x+y)2 (x+y)2 is not equal to x2+y2x2+y2
First step in multiplying (x+y)2 and (x+y)2 is to expand (x+y)2 twice.
(x+y)2 (x+y)2 = (x2+y2+2xy) (x2+y2+2xy)
Then start multiplying both the expressions. First multiply x2 of the 1st expression with the whole of 2nd expression
Multiply Y2 of the 1st expression with the whole of 2nd expression
Multiply 2xy of the 1st expression with the whole of 2nd expression
Add all the results from step 2 to step 4
X4+x2y2+2x3y+x2y2+y4+2xy3+2x3y+2xy3+4x2y2
Combine the expressions x2y2 as highlighted below
X4+x2y2+2x3y+x2y2+y4+2xy3+2x3y+2xy3+4x2y2
Now you get the below result
X4+6x2y2+2x3y+y4+2xy3+2x3y+2xy3
Combine 2x3y and 2xy3 as highlighted below
X4+6x2y2+2x3y+y4+2xy3+2x3y+2xy3
Now you get the below result
X4+6x2y2+4x3y+4xy3+y4
So, this is clearly not equal to x2+y2x2+y2. By expanding (x+y)2 twice, we get the expressions x4 and y4. Time and again algebraic multiplication has helped in proving many hypothesis wrong.
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