For which value of r will (2r + 5)x − 10 not remain a linear polynomial?

Correct answer: r = −5/2

The governing concept is the degree of a polynomial. An expression in x is linear when the coefficient of x is nonzero, because then its degree is 1. In (2r+5)x−10, the coefficient of x is 2r+5. To make the expression no longer linear, set that coefficient equal to zero: 2r+5=0, so 2r=−5 and r=−5/2. Substitution gives 0x−10=−10, a constant polynomial of degree 0. Therefore option B is correct. For r=0, the coefficient is 5; for r=5, it is 15; and for r=5/2, it is 10, so each of those cases remains linear. The constant term −10 does not affect whether the x-term has degree one.