To determine which algebraic expression is a polynomial with a degree of 4, we need to examine each expression individually and identify the highest degree of [tex]\( x \)[/tex] in each polynomial.
1. [tex]\( 5x^4 + \sqrt{4x} \)[/tex]
- The term [tex]\( 5x^4 \)[/tex] has a degree of 4.
- The term [tex]\( \sqrt{4x} \)[/tex] is not a polynomial term since it can be rewritten as [tex]\( 2\sqrt{x} \)[/tex], and [tex]\( \sqrt{x} \)[/tex] is [tex]\( x^{1/2} \)[/tex].
Since [tex]\( 5x^4 + \sqrt{4x} \)[/tex] includes the term [tex]\( \sqrt{4x} \)[/tex], it is not a polynomial.
2. [tex]\( x^5 - 6x^4 + 14x^3 + x^2 \)[/tex]
- The term [tex]\( x^5 \)[/tex] has a degree of 5.
- The term [tex]\( -6x^4 \)[/tex] has a degree of 4.
- The term [tex]\( 14x^3 \)[/tex] has a degree of 3.
- The term [tex]\( x^2 \)[/tex] has a degree of 2.
The highest degree term in [tex]\( x^5 - 6x^4 + 14x^3 + x^2 \)[/tex] is [tex]\( x^5 \)[/tex], which has a degree of 5.
3. [tex]\( 9x^4 - x^3 - \frac{x}{5} \)[/tex]
- The term [tex]\( 9x^4 \)[/tex] has a degree of 4.
- The term [tex]\( -x^3 \)[/tex] has a degree of 3.
- The term [tex]\( -\frac{x}{5} \)[/tex] or [tex]\( -\frac{1}{5}x \)[/tex] has a degree of 1.
The highest degree term in [tex]\( 9x^4 - x^3 - \frac{x}{5} \)[/tex] is [tex]\( 9x^4 \)[/tex], which has a degree of 4. Therefore, this is a polynomial of degree 4.
4. [tex]\( 2x^4 - 6x^4 + \frac{14}{x} \)[/tex]
- The term [tex]\( 2x^4 \)[/tex] has a degree of 4.
- The term [tex]\( -6x^4 \)[/tex] has a degree of 4.
- The term [tex]\( \frac{14}{x} \)[/tex] or [tex]\( 14x^{-1} \)[/tex] is not a polynomial term, as [tex]\( x \)[/tex] is in the denominator.
Combining like terms [tex]\( 2x^4 - 6x^4 \)[/tex] gives [tex]\( -4x^4 \)[/tex],
which has a degree of 4, but the presence of [tex]\( \frac{14}{x} \)[/tex] makes it not a polynomial.
Conclusion:
The only algebraic expression which is a polynomial with a degree of 4 is:
[tex]\[ 9x^4 - x^3 - \frac{x}{5} \][/tex]
