If \(f(x) = 2x^7 + 3x - 5 \), which of the following is a factor of \(f(x)\)?

Answer: (x-1) is a factor.

Reason: Let g(x) = f(x^2) = 2x^7 + 3x - 5. Evaluate g at x = 1:

g(1) = 2(1)^7 + 3(1) - 5 = 0, so f(1^2) = f(1) = 0.

Because f(1) = 0, the factor (x-1) divides f(x).

Why the other choices fail to follow from the given information:

• If (x+1) were a factor of f(x), then f(x^2) would be divisible by x^2+1. But g(i) = 2i^7 + 3i - 5 = i - 5 ≠ 0, so x^2+1 does not divide g(x).

• If (2x-5) were a factor of f(x), then f(x^2) would be divisible by 2x^2 - 5. Evaluating g at x = √(5/2) (≈ 1.5811) does not give zero, so 2x^2 - 5 does not divide g(x).

• If (x^3+8) were a factor of f(x), then f(x^2) would be divisible by x^6+8. There is no evidence that x^6+8 divides 2x^7+3x-5, so this choice is not supported.