Question

select the correct answer. let p be the set of polynomials. let a, b, c, and d be elements of p such that b, c, and d are nonzero elements. which statement about this quotient must be true? \\(\boldsymbol{\frac{a}{b} div \frac{c}{d}}\\) \\(\circ\\) the quotient is a rational expression. \\(\circ\\) the quotient is a polynomial. \\(\circ\\) the quotient is a rational number. \\(\circ\\) the quotient is an integer.

Explanation:

Step1: Rewrite the division

Dividing by a fraction is multiplying by its reciprocal:
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$

Step2: Analyze the result

A rational expression is defined as a ratio of two polynomials where the denominator is non-zero. Here, $a,b,c,d$ are polynomials, $b,c,d$ are non-zero, so $bc$ is non-zero. $\frac{ad}{bc}$ fits the definition of a rational expression. It is not guaranteed to be a polynomial (e.g., if $a=1, b=x, c=1, d=1$, the result is $\frac{1}{x}$, not a polynomial), nor a rational number or integer (since the elements are polynomials, not necessarily constants).

Answer:

The quotient is a rational expression.