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Find the coefficent of x^(4) in the pro...

Find the coefficent of ` x^(4)` in the product `(1+2x)^(4) xx (2-x)^(5).`

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Using the binomial expansion, we get
`(1+2x)^(4) = .^(4) C _(0) + .^(4) C _(1) ( 2x) + .^(4) C _(2) (2x)^(2) + .^(4) C _(3) ( 2x) ^ (3) + .^(4) C _(4) ( 2x) ^ (4)`
`=1 + 8x + 24x^(2) + 32x^(3) + 16x^(4)`
and `( 2-x) ^ (5) = 2^(5) - . ^ (5) C _(1) (2^(4))x + .^(5) C _ (2)(2^(3))x^(2) - .^(5) C _(3) ( 2^(2))x^(3)`
`=.^(5) C _(4) ( 2) x^(4) - .^(5) C _ (5) x ^ (5)`
`= 32 - 80x + 80x^(2) - 40x^(3) + 10x^(4) - x^(5)`
` :. (1+2x)^(4) xx (2-x)^(5)`
` =(1+ 8x + 24x^(2) + 32x^(3) + 16x^(4))`
`xx ( 32 - 80x + 80x^(2) - 40x^(3) + 10x^(4) - x^(5)).`
Sum of the terms containing `x^(4)` in the given product
`(1 xx 10x^(4)) + 8x (-40x^(3)) + (24x^(2)) (80x^(2))`
` ( 32 x^(3) ) (-80x) + ( 16x^(4) xx 32 ) `
`= 10^(x^(4)) - 320 x^(4) + 1920x^(4) - 2560x^(4) + 512 x^(2) `
`= (10- 320 + 1920 - 2560 + 512) x^(4) = - 438x^(4)`.
Hence , the coefficient of `x^(4)` in the given product is `-438`.
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RS AGGARWAL-BINOMIAL THEOREM-EXERCISE 10B
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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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