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Expand (1+ x+ x^(2))^(3) using binomial...

Expand `(1+ x+ x^(2))^(3)` using binomial expansion.

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Put `1+x = y`. Then ,
`(1 + x+ x^(2)) ^(3) = ( y + x^(2) ) ^ ( 3) `
`=.^(3) C _ (0)y^(3) + .^(3) y C_(1)y^(2) x^(2) + .^(3) C_(2)y(x^(2))^(2)+ .^(3) C _(3) (x^(2))^(3)`
`= y^(3) + 3y^(2) x^(2) + 3yx^(4) + x^(6) `
`(1+x)^(3) + 3x^(2) (1+x)^(2) + 3x^(4) ( 1+x) +x^(6) " " [:' y= (1+x) ]`
`= (1+.^(3)C_(1)x + ^(3)C_(3)x^(3)) + 3x^(2) ( 1+ 2x + x^(2)) + 3x^(4) ( 1+x) + x^(6)`
`=(1+ 3x + 3x^(2) + x^(3)) + ( 3x^(2) + 6x^(3) + 3x^(4)) + ( 3x^(4) + 3x^(5)) + x^(6)`
`=x^(6) + 3x^(5) + 6x^(4) = 7x^(3) + 6x^(2) + 3x + 1 .`
Hence,`(1+x+x^(2))^(3) = x^(6) + 3x^(5) + 6x^(4) + 7x^(3) + 6x^(2) + 3x + 1 .`
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RS AGGARWAL-BINOMIAL THEOREM-EXERCISE 10B
  1. Expand (1+ x+ x^(2))^(3) using binomial expansion.

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  2. Show that the term independent of x in the expansion of (x-1/x)^(10) ...

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  3. If the coefficents of x^(2) and x^(3) in the expansion of (3+px)^(9)...

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  4. Show that coefficient of x^(-3) in the expansion of (x-1/x)^(11) is -3...

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  5. Show that the middle term in the expansion of ((2x^2)/3+3/(2x)^(2))^(1...

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  6. Show that the coefficient of x^(4) in the expansion of (x/2-3/x^(2))^(...

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  7. Prove that there is no term involving x^(6) is the expansion of (2x^(2...

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  8. Show that the coefficient of x^(4) in the expansion of (1+2x+x^(2))^(5...

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  9. Write the number of terms in the expansion of (sqrt(2)+1)^(5)+ (sqrt(...

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  10. Which term is independent of x in the expansion of (x-1/(3x^(2)))^(9)?

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  11. Write the coefficient of the middle term in the expansion of (1+x)^(2...

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  12. Write the coefficient of x^(7) y^(2) in the expansion of (x+2y)^(9).

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  13. If the coefficent of (r-5)th and (2r-1)th terms in the expansion of (...

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  14. Write the 4th term form the end in the expansion of (3/(x^(2))-x^(3)/6...

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  15. Find the coefficient of x^n in the expansion of (1+x)(1+x)^ndot

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  16. In the binomial expansion of (a+b)^n , coefficients of the fourth and ...

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  17. Find a positive value of m for which the coefficient of x^2 in the ex...

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