Advertisements
Advertisements
Question
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
Advertisements
Solution
Using binomial theorem, the given expression (3x* -2ax +3a* ) can be expanded
[3x2 - a (2x - 3a)]3
= 3C0 (3x2 - 2ax)3 + 3C1(3x2 - 2ax)2 (3a2)+ 3C2(3x2 - 2ax) (3a2)2 + 3C3(3a2)3
= (3x2 - 2ax)3 + 3(9x4 - 12ax3 + 4a2x2)(3a2)+3(3x2 - 2ax)(9a4) + 27a6
= (3x2 - 2ax)3 + 81a2x4 - 108a3x3 + 36a4x2 + 81a4x2 - 54a5x + 27a6
= (3x2 - 2ax)3 + 81a2x4 - 108a3x3 + 117a4x2 - 54a5x + 27a6
Again, by using binomial theorem, we obtain
(3x2 - 2ax)3
= 3C0 (3X2)3 - 3C1 (3X2)2 (2ax) + 3C2 (3X2)(2ax)2 - 3C3 (2ax)3
= 27x6 - 3(9x4) (2ax) + 3 (3x2) (4a2x2) -8a3x3
= 27x6 - 54ax5 + 36a2x4 - 8a3x3
From (1) and (2), we obtain
(3x2 - 2ax + 3a2)3
= 27x6 - 54ax5 + 36a2 x4 - 8a3x3 + 81a2x4 - 108a3x3 + 117a4 x2 - 54a5x + 27a6
= 27x6 - 54ax5 + 117a2 x4 - 116a3 x3 + 117a4 x2 - 54a5x + 27a6
APPEARS IN
RELATED QUESTIONS
Expand the expression: (1– 2x)5
Expand the expression: (2x – 3)6
Expand the expression: `(x/3 + 1/x)^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
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Prove that `sum_(r-0)^n 3^r ""^nC_r = 4^n`
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 an approximation of (0.99)5 using the first three terms of its expansion.
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
Find the coefficient of x15 in the expansion of (x – x2)10.
Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.
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.
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O2 – E2 = (x2 – a2)n
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then ______.
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.
Let the coefficients of x–1 and x–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn2 = 15Cr, 2r, then the value of r is equal to ______.
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
