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 and d are nonzero elements of p. which statement about this sum
$\frac{a}{b} + \frac{c}{d}$
is true?
a. the sum is a rational number.
b. the sum is a rational expression.
c. the sum is an integer.
d. the sum is a polynomial.

Explanation:

Step1: Recall Definitions

A rational expression is a fraction where both the numerator and the denominator are polynomials (and the denominator is non - zero). Here, \(a,b,c,d\) are polynomials (since they are elements of the set of polynomials \(P\)) and \(b
eq0\), \(d
eq0\). So \(\frac{a}{b}\) and \(\frac{c}{d}\) are rational expressions.

Step2: Analyze the Sum of Rational Expressions

When we add two rational expressions \(\frac{a}{b}+\frac{c}{d}\), we find a common denominator (which is \(bd\), and since \(b\) and \(d\) are non - zero polynomials, \(bd
eq0\)). The sum is \(\frac{ad + bc}{bd}\). The numerator \(ad+bc\) is a polynomial (because the sum and product of polynomials are polynomials) and the denominator \(bd\) is a non - zero polynomial. So \(\frac{ad + bc}{bd}\) is also a rational expression.

Step3: Eliminate Other Options

• Option A: The sum is not necessarily a rational number. For example, if \(a = x\), \(b = 1\), \(c=x\), \(d = 1\) (where \(x\) is a variable), then \(\frac{a}{b}+\frac{c}{d}=x + x=2x\), which is not a rational number (it's a polynomial, but more generally, if the polynomials have variables, the sum is not a number).
• Option C: The sum is not necessarily an integer. Using the same example as above, \(2x\) is not an integer (unless \(x\) takes integer values, but \(x\) is a polynomial variable).
• Option D: The sum is not necessarily a polynomial. For example, if \(a = 1\), \(b=x\), \(c = 1\), \(d=x\), then \(\frac{a}{b}+\frac{c}{d}=\frac{1}{x}+\frac{1}{x}=\frac{2}{x}\), which is not a polynomial (polynomials have non - negative integer exponents, and here the exponent of \(x\) is \(- 1\)).

Answer:

B. The sum is a rational expression.