Which algebraic expression is a polynomial with a degree of $4$?
A. $5 x^4+\sqrt{4 x}$
B. $x^5-6 x^4+14 x^3+x^2$
C. $9 x^4-x^3-\frac{x}{5}$
D. $2 x^4-6 x^4+\frac{14}{x}$
Answer (2)
Identify polynomials: Check for non-negative integer exponents.
Determine the degree: Find the highest power of the variable.
Expression 1 has a fractional exponent, so it is not a polynomial.
Expression 2 is a polynomial of degree 5.
Expression 3 is a polynomial of degree 4.
Expression 4 has a negative exponent, so it is not a polynomial.
The polynomial with a degree of 4 is 9 x 4 − x 3 − 5 x .
Explanation
Understanding Polynomials We need to identify the algebraic expression that is a polynomial with a degree of 4. A polynomial expression contains variables and coefficients, and it only involves addition, subtraction, and non-negative integer exponents. The degree of the polynomial is the highest power of the variable in the expression.
Expression 1 Let's examine each expression:
5 x 4 + 4 x = 5 x 4 + 2 x 2 1 . This is not a polynomial because it contains a fractional exponent ( 2 1 ).
Expression 2
x 5 − 6 x 4 + 14 x 3 + x 2 . This is a polynomial. The highest power of x is 5, so the degree of this polynomial is 5.
Expression 3
9 x 4 − x 3 − 5 x = 9 x 4 − x 3 − 5 1 x . This is a polynomial. The highest power of x is 4, so the degree of this polynomial is 4.
Expression 4
2 x 4 − 6 x 4 + x 14 = 2 x 4 − 6 x 4 + 14 x − 1 . This is not a polynomial because it contains a negative exponent ( − 1 ).
Conclusion Therefore, the algebraic expression that is a polynomial with a degree of 4 is 9 x 4 − x 3 − 5 x .
Examples
Polynomials are used to model curves and relationships in various fields such as physics, engineering, computer graphics, and economics. For example, the trajectory of a projectile can be modeled using a quadratic polynomial, and the growth of a population can be modeled using an exponential function, which can be approximated by a polynomial. Understanding polynomials helps in analyzing and predicting the behavior of these systems.
The polynomial expression with a degree of 4 is option C: 9 x 4 − x 3 − 5 x . This expression has the highest exponent of 4, qualifying it as a degree 4 polynomial. Options A and D are not polynomials due to fractional and negative exponents, respectively, and option B has a degree of 5.
;
