Advertisements
Advertisements
Question
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
Advertisements
Solution
`(1 + x/2 - 2/x)^4 = [(1 + x/2) - 2/x]^4`
= `(1 + x/4)^4 + ^4C_1 (1 + x/2)^3 (-2/x) + ^4C_2 (1 + x/2)^2 (-2/x)^2 + ^4C_3 (1 + x/2) (-2/x)^3 + ^4C_4 (-2/x)^4`
= `(1 + x/2)^4 + 4(1 + x/2)^3 (-2/x) + 6 (1 + x/2)^2 (4/x^2) + 4 (1 + x/2) (- 8/x^3) + (16/x^4)`
= `(1 + x/2)^4 , (1 + x/2)^3 , (1 + x/2)^2` on spreading
= `(1 + x/2 - 2/x)^4 = (1 + 4. x/2 + 6 x^2/4 + 4. x^3/8 + x^4/16) - 8/x (1 +3 . x/2 + 3. x^2/4 + x^3/8) + 24/x^2 (1 + x + x^2/4) - 32/x^3 (1 + x/2) + 16/x^4`
= `(1 + 2x + 3/2 x^2 + 1/2 x^3 + x^4/16) - 8/x(1 + 3/2 x + 3/4 x^2 + x^3/8) + 24/x^2 (1 + x + x^2/4) - 32/x^3 (1 + x/2) + 16/x^4`
= `(1 + 2x + 3/2 x^2 + 1/2 x^3 + x^4/16) - (8/x + 12 + 6x + x^2) + (24/x^2 + 24/x + 6) - (32/x^3 + 16/x^2) + 16/x^4`
= `x^4/16 + x^3/2 + (3/2 - 1)x^2 + (2 -6)x + (1 - 12 +6) + (- 8 + 24) 1/x + (24 -16) 1/x^2 - 32/x^3 + 16/x^4`
= `x^4/16 + x^3/2 + x^2/2 - 4x -5 + 16/x + 8/x^2 - 32/x^3 + 16/x^4`
APPEARS IN
RELATED QUESTIONS
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Using Binomial Theorem, evaluate the following:
(96)3
Using Binomial Theorem, evaluate of the following:
(102)5
Using binomial theorem, evaluate f the following:
(101)4
Using binomial theorem, evaluate the following:
(99)5
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively.
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find an approximation of (0.99)5 using the first three terms of its expansion.
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Expand the following (1 – x + x2)4
Find the 4th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
If n is a positive integer, find the coefficient of x–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
The number of terms in the expansion of (a + b + c)n, where n ∈ N is ______.
If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
Find the coefficient of x15 in the expansion of (x – x2)10.
If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
The number of terms in the expansion of (x + y + z)n ______.
The coefficient of a–6b4 in the expansion of `(1/a - (2b)/3)^10` is ______.
Number of terms in the expansion of (a + b)n where n ∈ N is one less than the power n.
The positive integer just greater than (1 + 0.0001)10000 is ______.
