To determine which algebraic expression is a polynomial with a degree of 4, we need to carefully analyze each given expression.
### 1. [tex]\( 5x^4 + \sqrt{4x} \)[/tex]
- [tex]\( 5x^4 \)[/tex] is a polynomial term with a degree of 4.
- [tex]\( \sqrt{4x} \)[/tex] can be rewritten as [tex]\( 2\sqrt{x} \)[/tex]. The term [tex]\( \sqrt{x} \)[/tex] can be written as [tex]\( x^{1/2} \)[/tex], which is not a polynomial term because its exponent is not a non-negative integer.
Therefore, this expression is not a polynomial.
### 2. [tex]\( x^5 - 6x^4 + 14x^3 + x^2 \)[/tex]
The expression consists of polynomial terms:
- [tex]\( x^5 \)[/tex] with a degree of 5,
- [tex]\( -6x^4 \)[/tex] with a degree of 4,
- [tex]\( 14x^3 \)[/tex] with a degree of 3,
- [tex]\( x^2 \)[/tex] with a degree of 2.
The highest degree term here is [tex]\( x^5 \)[/tex], which means the overall degree of this polynomial is 5, not 4.
### 3. [tex]\( 9x^4 - x^3 - \frac{x}{5} \)[/tex]
The expression consists of polynomial terms:
- [tex]\( 9x^4 \)[/tex] with a degree of 4,
- [tex]\( -x^3 \)[/tex] with a degree of 3,
- [tex]\( -\frac{x}{5} \)[/tex], which is [tex]\( -\frac{1}{5}x \)[/tex] with a degree of 1.
The highest degree term here is [tex]\( 9x^4 \)[/tex], so the overall degree of this polynomial is 4. Thus, this expression is a polynomial of degree 4.
### 4. [tex]\( 2x^4 - 6x^4 + \frac{14}{x} \)[/tex]
- [tex]\( 2x^4 \)[/tex] is a polynomial term with a degree of 4,
- [tex]\( -6x^4 \)[/tex] is also a polynomial term with a degree of 4,
- [tex]\( \frac{14}{x} \)[/tex] can be rewritten as [tex]\( 14x^{-1} \)[/tex], which is not a polynomial term because its exponent is negative.
Thus, this expression is not a polynomial.
### Conclusion
The only expression that is a polynomial with a degree of 4 is:
[tex]\[ 9x^4 - x^3 - \frac{x}{5} \][/tex]
So, the expression [tex]\( 9x^4 - x^3 - \frac{x}{5} \)[/tex] is the polynomial with a degree of 4.
