- Rational expressions are defined as the ratio of two polynomials, \(\frac{P(x)}{Q(x)}\) with \(Q(x)
eq 0\).
- Multiplying two rational expressions \(\frac{a}{b}\) and \(\frac{c}{d}\) results in \(\frac{ac}{bd}\), where \(ac\) and \(bd\) are both polynomials.
- Since the set of polynomials is closed under multiplication, \(ac\) and \(bd\) are polynomials, and since \(b, d
eq 0\), it follows that \(bd
eq 0\).
- Therefore, the product is always a rational expression, whereas other classifications are not guaranteed.
The correct statement is: \(\boxed{\text{The product is a rational expression.}}\)
Explanation
- Analyzing the definition of a rational expression
To identify the correct statement about the product \(\frac{a}{b} \cdot \frac{c}{d}\), we first recall the definition of a rational expression. A rational expression is defined as a fraction of the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not the zero polynomial. In our case, \(a, b, c,\) and \(d\) are given as polynomials, with \(b
eq 0\) and \(d
eq 0\).
- Evaluating the product of two rational expressions
The product of two rational expressions \(\frac{a}{b}\) and \(\frac{c}{d}\) is given by the formula:
\[ \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \]
By the closure property of polynomials under multiplication, since \(a\) and \(c\) are polynomials, their product \(ac\) is also a polynomial. Similarly, because \(b\) and \(d\) are polynomials, their product \(bd\) is a polynomial. Since polynomials form an integral domain, the product of two non-zero polynomials is guaranteed to be non-zero. Thus, \(bd
eq 0\). The expression \(\frac{ac}{bd}\) satisfies all the criteria for being a rational expression.
- Comparing and verifying the options
We now evaluate the other options:
- "The product is a polynomial": This is not always true. For example, if \(a=1, b=x, c=1, d=1\), then \(\frac{1}{x} \cdot 1 = \frac{1}{x}\), which is not a polynomial.
- "The product is an integer": This is not always true, as the variable \(x\) can take any value, and the resulting expression depends on \(x\).
- "The product is a rational number": A rational number is a ratio of two integers, whereas our result is a ratio of two polynomials, which is generally not a constant number.
Thus, the only statement that must be true is that the product is a rational expression.
Examples
In computer algebra systems or symbolic mathematics software, rational expressions are handled as distinct data types. Just as numerical fractions allow us to represent any value between integers, rational expressions allow us to represent any function formed by dividing two polynomials. When you multiply two such functions together, the software ensures the result remains within the domain of rational functions. This is crucial for simplifying complex algebraic circuits, modeling signals in engineering, or performing calculus operations on rational functions where the resulting form must remain mathematically consistent.