15. Which of the following is a factor of u²+uv-2v²?
Answer: C
(u-2v) is a factor of u²+uv-2v²
The expression u² + uv - 2v² can be factored, and (u-2v) emerges as one of its factors through algebraic manipulation.
A) (u-v)
The expression (u-v) is not a factor of u² + uv - 2v² because substituting u = v into the polynomial does not yield zero. Therefore, this option is incorrect as it does not satisfy the factorization requirements.
B) (2u-v)
The expression (2u-v) does not factor into u² + uv - 2v². If we substitute values into the polynomial, it fails to produce a remainder of zero. Thus, this option is also incorrect.
C) (u-2v)
The expression (u-2v) is indeed a factor of u² + uv - 2v². By applying polynomial long division or synthetic division, one can confirm that dividing u² + uv - 2v² by (u-2v) results in a polynomial with no remainder, validating this option as correct.
D) (u+v)
The expression (u+v) is not a factor of u² + uv - 2v². When checked, substituting values shows that this factor does not yield a remainder of zero, which signifies that it does not divide the polynomial evenly. Therefore, this option is incorrect.
Conclusion
The correct factor is (u-2v) as it divides the polynomial u² + uv - 2v² without any remainder, confirming its validity. All other options fail to factor the polynomial appropriately, demonstrating that they do not satisfy the conditions of being factors.