Let p(x)=x^{4}+x^{2}+1, q(x)=x^{4}-2x^{3}+3x^{2}-2x+1. If GCD of p(x) and q(x) is x^{2}-x+1, then what is their LCM?
- A. (x^{2}+x+1)(x^{2}-x+1)^{2} ✓
- B. (x^{4}+x^{2}+1)^{2}(x^{2}-x+1)
- C. (x^{4}+x^{2}+1)(x^{2}+x+1)^{2}
- D. (x^{4}+x^{2}+1)(x^{2}-x+1)^{2}
Correct Answer: A. (x^{2}+x+1)(x^{2}-x+1)^{2}
Explanation
Factoring yields p(x) = (x^2+x+1)(x^2-x+1) and q(x) = (x^2-x+1)^2. The LCM is the product of the highest powers of all prime factors, which evaluates to (x^2+x+1)(x^2-x+1)^2.
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