In the binomial expansion of (2x - 1/x)^8, what is the term independent of x (the constant term)?
Answer and explanation
Correct answer: C. 1120
- Step 1: Write general term: T_(r+1) = C(8, r) (2x)^(8-r) (-1/x)^r.
- Step 2: Separate coefficients and powers of x: C(8, r) 2^(8-r) (-1)^r * x^(8 - 2r).
- Step 3: For the term independent of x, set power to 0: 8 - 2r = 0 => 2r = 8 => r = 4.
- Step 4: Compute coefficient for r = 4:
- Step 5: C(8, 4) = 8! / (4! * 4!) = 70.
- Step 6: 2^(8 - 4) = 2^4 = 16.
- Step 7: (-1)^4 = 1.
- Step 8: Multiply: 70 16 1 = 1120.
Answer
1120.
Why not the others: '-1120' makes a sign error on (-1)^4; '1792' calculates C(8, 3) * 2^5; '-1792' combines power and sign errors.
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