Is it correct to treat \(4x^2+\sqrt{x^2}+1\) as a polynomial in (x)?

Correct answer: No, because \(\sqrt{x^2}=|x|\)

No. For real \(x\), \(\sqrt{x^2}=|x|\), not always \(x\). Thus the expression becomes \(4x^2+|x|+1\). Since \(|x|\) cannot be written as a term having a non-negative integral power of \(x\), the expression is not a polynomial in \(x\). Option A is true only when \(x\geq0\). Exam tip: whenever you see \(\sqrt{x^2}\), write \(|x|\) first, not simply \(x\).