Using the Factor Theorem, substituting x = k into the expression x - 29nk² allows solving for n. What does this imply?

• A. The value of n can be determined by substituting x = k into the expression and solving for n.
• B. The value of n is arbitrary.
• C. n must be zero.
• D. n equals k.

Answer: (A)

By using the Factor Theorem, you enforce that the expression has a root at x = k. That means you substitute x = k into the polynomial and set the result equal to zero, then solve for any parameters.

For the expression x - 29 n k^2, substituting x = k gives k - 29 n k^2. Setting this to zero yields k - 29 n k^2 = 0. Factor out k: k(1 - 29 n k) = 0. If k ≠ 0, then 1 - 29 n k = 0, so n = 1/(29 k). If k = 0, the equation becomes 0 = 0 and places no restriction on n, so n is not determined in that degenerate case.

So this substitution method shows you can determine the value of n from the equation, provided k is not zero.