Which statement is true about a rational expression?

Which statement is true about a rational expression?

• A. The numerator and denominator are polynomials, and the denominator cannot be zero.
• B. The numerator is always a monomial and the denominator is a constant.
• C. Rational expressions must be whole numbers.
• D. Rational expressions are always polynomials.

Answer: (A)

A rational expression is a ratio of two polynomials, with the denominator not equal to zero. Because both parts are polynomials, you’re looking at the form a(x)/b(x) where b(x) ≠ 0 for the values of x you’re considering. That’s why the statement is true: the numerator and denominator are polynomials, and you must exclude x-values that make the denominator zero, since division by zero isn’t defined. The numerator doesn’t have to be a monomial; it can be any polynomial. The denominator also doesn’t have to be a constant; it can be any polynomial as well. Rational expressions can evaluate to non-integer values, so they aren’t restricted to whole numbers. And they aren’t always polynomials themselves—the ratio of polynomials is, in general, a rational function, which is not a polynomial unless the denominator is a nonzero constant or the fraction simplifies to a polynomial.

A rational expression is a ratio of two polynomials, with the denominator not equal to zero. Because both parts are polynomials, you’re looking at the form a(x)/b(x) where b(x) ≠ 0 for the values of x you’re considering. That’s why the statement is true: the numerator and denominator are polynomials, and you must exclude x-values that make the denominator zero, since division by zero isn’t defined.

The numerator doesn’t have to be a monomial; it can be any polynomial. The denominator also doesn’t have to be a constant; it can be any polynomial as well. Rational expressions can evaluate to non-integer values, so they aren’t restricted to whole numbers. And they aren’t always polynomials themselves—the ratio of polynomials is, in general, a rational function, which is not a polynomial unless the denominator is a nonzero constant or the fraction simplifies to a polynomial.