When will ((u-5)x^3+(u-5)x^2+7) become a constant polynomial?

Correct answer: When (u=5)

The direct answer is A: the expression becomes a constant polynomial when \\(u=5\\). A constant polynomial must contain no variable term with a nonzero coefficient. Here the coefficients of both variable terms \\(x^3\\) and \\(x^2\\) are \\(u-5\\). Set that coefficient equal to zero: \\(u-5=0\\), so \\(u=5\\). Substitution gives \\(0x^3+0x^2+7=7\\), which is a constant polynomial. Option A is correct. Option B gives \\(u=-5\\), so \\(u-5=-10\\); the expression still contains nonzero \\(x^3\\) and \\(x^2\\) terms. Option C gives \\(u=0\\), so the common coefficient is \\(-5\\), and the variable terms remain. Option D is wrong because a suitable value does exist, namely 5. It is enough to make the common coefficient zero because the same coefficient controls both variable terms. The remaining number 7 is allowed as a constant. Exam cue: to make a polynomial constant, set every coefficient of a positive-power variable term equal to zero.