High School

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Which algebraic expression is a polynomial with a degree of 4?
Responses:
1) 5x^4 + √(4x)
2) x^5 - 6x^4 + 14x^3 + x^2
3) 9x^4 - x^3 - x/5
4) 2x^4 - 6x^4 + 14/x

Answer :

To solve this problem, we need to understand what a polynomial is and how to determine its degree.

A polynomial is an expression composed of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents of variables. The degree of a polynomial is the highest power of the variable present in the expression.

Let's assess each option to determine which is a polynomial with a degree of 4:

  1. [tex]5x^4 + \sqrt{4x}[/tex]

    • This expression includes a square root term, [tex]\sqrt{4x}[/tex], which is not a valid term in a polynomial. Therefore, this is not a polynomial with a degree of 4.

  2. [tex]x^5 - 6x^4 + 14x^3 + x^2[/tex]

    • This is indeed a polynomial. However, the highest degree term is [tex]x^5[/tex], giving it a degree of 5. So, this is not a polynomial with a degree of 4.

  3. [tex]9x^4 - x^3 - \frac{x}{5}[/tex]

    • This expression is a valid polynomial. The highest degree term is [tex]9x^4[/tex], which means the degree of the polynomial is 4. This fits the criteria.

  4. [tex]2x^4 - 6x^4 + \frac{14}{x}[/tex]

    • This expression includes a term, [tex]\frac{14}{x}[/tex], which is not allowed in polynomials because it has a negative exponent. Therefore, this is not a polynomial with a degree of 4.

After evaluating all options, the correct choice that is a polynomial with a degree of 4 is option 3: [tex]9x^4 - x^3 - \frac{x}{5}[/tex].

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