Question

Find the number of terms in the expansion of (3x+y)^8-(3x-y)^8

Answer 1

(3x + y)^8 has 9 terms. So has (3x - y)^8.

So there are a total of 18 terms. But some terms will be advanced and deducted as both negative and positive terms of some terms are there in the expansion.

Let's see it.

In the expansion of (3x - y)^8, the terms with coefficients 8C1, 8C3, 8C5 and 8C7 have negative signs. But these will be positive as there is difference of (3x + y)^8 and (3x - y)^8 in the question, and the positive terms in the expansion of (3x - y)^8 will be negative.

The positive terms in the expansion of (3x - y)^8 are those with coefficients 8C0, 8C2, 8C4, 8C6 and 8C8. So there are 5 ones.

The twice of this 5 is to be subtracted from the total 18, thus we get the answer 18 - 10 = 8.

This remaining 8 terms are actually pairs of 4 terms. So we have to divide this 8 by 2, and thus we get 4. This is the answer.

So there are 4 terms in the expansion of (3x + y)^8 - (3x - y)^8.

Let's check.

=> (3x + y)^8 - (3x - y)^8

=> (6561x^8 + 17496x^7y + 20412x^6y^2 + 13608x^5y^3 + 5670x^4y^4 + 1512x^3y^5 + 252x^2y^6 + 24xy^7 + y^8) - (6561x^8 - 17496x^7y + 20412x^6y^2 - 13608x^5y^3 + 5670x^4y^4 - 1512x^3y^5 + 252x^2y^6 - 24xy^7 + y^8)

=> 6561x^8 + 17496x^7y + 20412x^6y^2 + 13608x^5y^3 + 5670x^4y^4 + 1512x^3y^5 + 252x^2y^6 + 24xy^7 + y^8 - 6561x^8 + 17496x^7y - 20412x^6y^2 + 13608x^5y^3 - 5670x^4y^4 + 1512x^3y^5 - 252x^2y^6 + 24xy^7 - y^8

=> 34992x^7y + 27216x^5y^3 + 3024x^3y^5 + 48xy^7

Hence approved!!!

Hope this helps. Please ask me if you have ANY doubt in my answer because I was not able to explain my answer accordingly as I don't have much time.

Thank you. :-))

$\mathfrak{\#adithyasajeevan}$