Question

question 4 of 10
select the correct answer.
let ( p ) be the set of polynomials. let ( a ), ( b ), ( c ), and ( d ) be elements of ( p ) such that ( b ) and ( d ) are non - zero elements in ( p ). which statement about this difference is true?
( \frac{a}{b}-\frac{c}{d} )

• the difference is a rational number.
• the difference is a rational expression.
• the difference is a polynomial.
• the difference is an integer.

Explanation:

Step1: Recall definitions

A rational expression is a fraction where the numerator and denominator are polynomials (and the denominator is non - zero). Here, \(a,b,c,d\) are polynomials, and \(b
eq0\), \(d
eq0\).

Step2: Simplify the difference

We simplify \(\frac{a}{b}-\frac{c}{d}\) using the common denominator \(bd\). So \(\frac{a}{b}-\frac{c}{d}=\frac{ad - bc}{bd}\).
The numerator \(ad - bc\) is a polynomial (since the product and difference of polynomials are polynomials) and the denominator \(bd\) is a non - zero polynomial (because \(b\) and \(d\) are non - zero polynomials). So \(\frac{ad - bc}{bd}\) is a rational expression.
A rational number is a number of the form \(\frac{p}{q}\) where \(p,q\) are integers and \(q
eq0\), but here \(a,b,c,d\) are polynomials, not necessarily integers. A polynomial has no variable in the denominator, but our result has a polynomial in the denominator (unless \(bd = 1\), which is not guaranteed). An integer is a whole number, and our result is a fraction of polynomials, not necessarily an integer.

Answer:

The difference is a rational expression.