Question

(x+y)³ =


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Solve the identity !
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Answer 1

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By answer, I assume you mean the expanded form?

(x+y)3=(x+y)(x+y)(x+y)

taking the first two:

(x+y)(x+y)=x(x+y)+y(x+y)

=x2+xy+yx+y2

=x2+2xy+y2.

This gives us

(x+y)3=(x2+2xy+y2)(x+y)

this can be expanded out into:

x(x2+2xy+y2)+y(x2+2xy+y2)=x3+2x2y+xy2+yx2+2xy2+y3

this can be simplified into

x3+3x2y+3xy2+y3

so (x+y)3=x3+3x2y+3xy2+y3

Answer 2

Step-by-step explanation:

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Hey friend,

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A short nicer way to solve this is the Binomial theorem:

This seems difficult but its simple when you will try it.

If you are also finding it difficult to do combinatorics, Pascal's triangle is the solution for you:

On the left, the number are the exponent we have to raise our binomial to and on the right the numbers are the coefficients on each terms of the answer in simplified form.

Example - calculate (x+y)³.

Ans-

From Pascal's triangle, we get the coefficients:

1( )+3( )+3( )+1( )

Now, from the binomial theorem,

We get,

1x³+3x²y+3xy³+1y³

And if you notice a pattern in the exponents of x and y, You don't even need binomial theorem!

(exponent on x are decreasing from 3 to 0 and exponent on y are increasing from 0 to 3 term by term.)

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Hope this helped you !!