If (p(x)=mx+n) and (p(1)=p(2)), when can (p(x)) be linear?

Correct answer: Never

For \(p(x)=mx+n\), we have \(p(1)=m+n\) and \(p(2)=2m+n\). Given \(p(1)=p(2)\), \(m+n=2m+n\), which gives \(m=0\). However, a linear polynomial must have a non-zero coefficient of \(x\). When \(m=0\), \(p(x)=n\) is a constant polynomial, not a linear polynomial. Hence, it can never be linear. Exam tip: always check that the coefficient of \(x\) is non-zero for a linear polynomial.