Answer :
To determine which of the given expressions are polynomials, we need to examine each expression independently and check whether it meets the criteria for being a polynomial. A polynomial is a mathematical expression involving a sum of powers of [tex]\(x\)[/tex] with non-negative integer exponents and real coefficients.
Let's analyze each expression one by one:
1. [tex]\(\frac{x}{5} + 2\)[/tex]
This expression can be rewritten as:
[tex]\[
\frac{1}{5} x + 2
\][/tex]
Here, [tex]\(\frac{1}{5}\)[/tex] and 2 are real coefficients, and [tex]\(x\)[/tex] is raised to the power of 1 (which is a non-negative integer). Therefore, this expression is a polynomial.
2. [tex]\(7 x^7 - x^5\)[/tex]
This expression consists of two terms: [tex]\(7 x^7\)[/tex] and [tex]\(- x^5\)[/tex]. Both terms have real coefficients (7 and -1) and [tex]\(x\)[/tex] is raised to the powers of 7 and 5 respectively (both non-negative integers). Therefore, this expression is a polynomial.
3. [tex]\(35 x^2\)[/tex]
This is a single-term expression where the coefficient is 35 (a real number) and [tex]\(x\)[/tex] is raised to the power of 2 (a non-negative integer). Therefore, this expression is a polynomial.
4. [tex]\(4 x + \sqrt{x} - 1\)[/tex]
Let's break this expression down:
- The term [tex]\(4 x\)[/tex] has a real coefficient 4 and [tex]\(x\)[/tex] is raised to the power of 1 (a non-negative integer).
- The term [tex]\(\sqrt{x}\)[/tex] can be rewritten as [tex]\(x^{1/2}\)[/tex], where 1/2 is not a non-negative integer. This violates the polynomial criteria.
- The term [tex]\(-1\)[/tex] is just a constant, which is fine for polynomials.
Because the term [tex]\(\sqrt{x}\)[/tex] (or [tex]\(x^{1/2}\)[/tex]) does not satisfy the polynomial form (non-negative integer exponent), this entire expression is not a polynomial.
Thus, the expressions that are polynomials are:
[tex]\[
\frac{x}{5} + 2, \quad 7 x^7 - x^5, \quad 35 x^2
\][/tex]
The only expression that is not a polynomial is:
[tex]\[
4 x + \sqrt{x} - 1
\][/tex]
Let's analyze each expression one by one:
1. [tex]\(\frac{x}{5} + 2\)[/tex]
This expression can be rewritten as:
[tex]\[
\frac{1}{5} x + 2
\][/tex]
Here, [tex]\(\frac{1}{5}\)[/tex] and 2 are real coefficients, and [tex]\(x\)[/tex] is raised to the power of 1 (which is a non-negative integer). Therefore, this expression is a polynomial.
2. [tex]\(7 x^7 - x^5\)[/tex]
This expression consists of two terms: [tex]\(7 x^7\)[/tex] and [tex]\(- x^5\)[/tex]. Both terms have real coefficients (7 and -1) and [tex]\(x\)[/tex] is raised to the powers of 7 and 5 respectively (both non-negative integers). Therefore, this expression is a polynomial.
3. [tex]\(35 x^2\)[/tex]
This is a single-term expression where the coefficient is 35 (a real number) and [tex]\(x\)[/tex] is raised to the power of 2 (a non-negative integer). Therefore, this expression is a polynomial.
4. [tex]\(4 x + \sqrt{x} - 1\)[/tex]
Let's break this expression down:
- The term [tex]\(4 x\)[/tex] has a real coefficient 4 and [tex]\(x\)[/tex] is raised to the power of 1 (a non-negative integer).
- The term [tex]\(\sqrt{x}\)[/tex] can be rewritten as [tex]\(x^{1/2}\)[/tex], where 1/2 is not a non-negative integer. This violates the polynomial criteria.
- The term [tex]\(-1\)[/tex] is just a constant, which is fine for polynomials.
Because the term [tex]\(\sqrt{x}\)[/tex] (or [tex]\(x^{1/2}\)[/tex]) does not satisfy the polynomial form (non-negative integer exponent), this entire expression is not a polynomial.
Thus, the expressions that are polynomials are:
[tex]\[
\frac{x}{5} + 2, \quad 7 x^7 - x^5, \quad 35 x^2
\][/tex]
The only expression that is not a polynomial is:
[tex]\[
4 x + \sqrt{x} - 1
\][/tex]
