Question

The constant term in the expansion of \( \left( x^3 + \frac{1}{x^2} \right)^{10} \) is:

• A. 210
• B. 240
• C. 140
• D. 120
• E. 320

Hint:

To find the constant term in a binomial expansion, solve for \( k \) where the exponent of \( x \) becomes zero.

Answer 1

The Correct Option is A

Using the binomial expansion formula, the general term is: \[ T_k = \binom{10}{k} (x^3)^{10-k} \left( \frac{1}{x^2} \right)^k \] \[ = \binom{10}{k} x^{3(10-k)} x^{-2k} \] \[ = \binom{10}{k} x^{30 - 5k} \] For the constant term, set \( 30 - 5k = 0 \): \[ 5k = 30 \implies k = 6 \] Substituting \( k = 6 \): \[ T_6 = \binom{10}{6} = 210 \] Thus, the constant term is 210.