The term independent of x in the expansion of
[ (x + 1)/(x^(2/3) - x^(1/3) + 1) - (x - 1)/(x - x^(1/2)) ]¹⁰, x ≠ 1, is equal to ____________.

Solution

Maths sir · 5 steps

Do not expand that monster. Simplify the bracket first and it becomes a plain binomial with two powers of x.

1. First fraction. Write x + 1 = (x^(1/3))³ + 1³, a sum of cubes, which factorises as (x^(1/3) + 1)(x^(2/3) - x^(1/3) + 1). The denominator cancels, leaving x^(1/3) + 1.
2. Second fraction. Write x - 1 = (√x - 1)(√x + 1) and x - √x = √x(√x - 1). Since x ≠ 1, the factor √x - 1 cancels, leaving (√x + 1)/√x = 1 + x^(-1/2).
3. General term. The bracket becomes (x^(1/3) + 1) - (1 + x^(-1/2)) = x^(1/3) - x^(-1/2). So we need the constant term of (x^(1/3) - x^(-1/2))¹⁰, and its general term is below.

   Tᵣ₊₁ = C(10, r)(-1)ʳ x^((10-r)/3 - r/2)

4. Power of x zero. Set (10-r)/3 - r/2 = 0. Then 20 - 2r = 3r, so r = 4. This term is C(10, 4)(-1)⁴ = 210.
5. So. The term independent of x is 210.

The idea to keep. When a binomial bracket looks ugly, look for sum of cubes or difference of squares hiding in fractional powers. Simplify first, then set the power of x in the general term to zero.

Answer from the NTA final answer key. Solution by PrepWiser.