Certainly! Let's analyze each option step-by-step to determine which one is a polynomial of degree 2:
(a) [tex]\(\sqrt{x} + 6x + 7\)[/tex]
To be classified as a polynomial, the expression should only contain non-negative integer exponents of [tex]\(x\)[/tex]. In this case, the term [tex]\(\sqrt{x}\)[/tex] is equivalent to [tex]\(x^{1/2}\)[/tex], which is not an integer exponent. Thus, this expression is not a polynomial of any degree.
(b) [tex]\(2x^3 + 3x^2 + 1\)[/tex]
A polynomial’s degree is determined by the highest power of [tex]\(x\)[/tex] in the expression. Here, the highest power of [tex]\(x\)[/tex] is 3 (from the term [tex]\(2x^3\)[/tex]). So this is a polynomial, but its degree is 3, not 2.
(c) [tex]\(x^2 + 2x + 1\)[/tex]
Let's check the terms:
- [tex]\(x^2\)[/tex] has a power of 2.
- [tex]\(2x\)[/tex] has a power of 1.
- The constant term 1 has a power of 0.
The highest power of [tex]\(x\)[/tex] is 2 (from the term [tex]\(x^2\)[/tex]). Therefore, this is indeed a polynomial of degree 2.
(d) None of these
This option means that none of the given expressions (a, b, c) would be a polynomial of degree 2. But, as we have already identified, option (c) is a polynomial of degree 2.
Based on this detailed analysis, the correct answer is:
(c) [tex]\(x^2 + 2x + 1\)[/tex]
