Which of the following options could be a denominator for a rational expression?
Option #1: x2−x0.5
Option #2: 8x14
Option #3: 5
All of the given options could be a denominator for a rational expression.
A rational expression is any algebraic expression that can be written as the quotient of two polynomials. All of the given options are polynomials or constants, hence can be the denominator of a rational expression. The only requirement is that the denominator cannot be equal to zero, because division by zero is undefined.
To determine which of the given options could be a denominator for a rational expression, we need to understand what a denominator is.
In a rational expression, the denominator represents the bottom part of the fraction, and it cannot be equal to zero.
Option #1: x^2 - x^0.5
This option includes the term x^0.5, which represents the square root of x. While this can be a part of a denominator, it cannot be the entire denominator. Therefore, option #1 cannot be a denominator for a rational expression.
Option #2: 8x^14
This option includes the term x^14, which is a power of x. Since this is just a power of x and does not involve any operations like square roots, it can be a valid denominator for a rational expression.
Option #3: 5
This option is a constant number, which can also be a valid denominator for a rational expression.
Therefore, both option #2 (8x^14) and option #3 (5) could be denominators for a rational expression.
To determine which of the options could be a denominator for a rational expression, we need to understand what a rational expression is and any restrictions on the variable.
A rational expression is a fraction in which both the numerator and denominator are polynomials. A polynomial is any expression consisting of variables, coefficients, and exponentiation, using addition, subtraction, and multiplication operations.
Let's analyze each option:
Option #1: x^2−x^0.5
This option includes a square root (√) in the denominator, which means it is not a polynomial. A rational expression should have a polynomial in the denominator, so option #1 is not a valid choice.
Option #2: 8x^14
This option includes a polynomial in the form of 8x^14. Since it meets the criteria of being a polynomial, option #2 could be a valid denominator for a rational expression.
Option #3: 5
This option is a constant term, which can be considered a polynomial of degree zero. Since it also meets the criteria of being a polynomial, option #3 could be a valid denominator for a rational expression.
In conclusion, both option #2 (8x^14) and option #3 (5) could be suitable denominators for a rational expression.