8.4 Rational Expressions
In the same way that one can extend the integers to the field of rational numbers by defining addition and multiplication on pairs of integers, one can extend the polynomial ring by defining addition and multiplication of pairs of polynomials and then defining a set of equivalence classes on this set.
Related Content Standards
- (HSA.APR.7) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a non-zero rational expression; add, subtract, multiply, and divide rational expressions.
Let \(F\) be a field and let \(F[x]\) be the integral domain of polynomials with coefficients from \(F\). We define an equivalence relation on the set of rational expressions of polynomials as \(\frac{p(x)}{q(x)}\) is equivalent to \(\frac{a(x)}{b(x)}\) if and only if \(p(x) \cdot b(x)=a(x)\cdot q(x)\).
Define \(F(x)\) to be the set of equivalence classes of quotients of polynomials \(\left[\frac{p(x)}{q(x)}\right]\) with \(q(x)\) not being the zero polynomial under this equivalence relation. We then proceed with dropping the equivalence class notation to simplify the notation. We define the following operations on \(F(x)\).
\[\frac{p(x)}{q(x)} + \frac{f(x)}{g(x)} = \frac{p(x)\cdot g(x) + q(x) \cdot f(x)}{q(x)\cdot g(x)} \quad \mbox{ and } \quad \frac{p(x)}{q(x)} \cdot \frac{f(x)}{g(x)} = \frac{p(x) \cdot f(x)}{q(x)\cdot g(x)}\]
We call the set \(F(x)\) the set of rational expressions with coefficients from \(F\) and we label the corresponding set with operations as \(\langle F(x),+,\cdot\rangle\), but usually refer to it as just \(F(x)\).
We see that the operations of addition and multiplication are well-defined because the polynomials form an integral domain and are therefore closed under addition and multiplication, and do not have any zero divisors. Thus if neither \(q(x)\) nor \(g(x)\) are the zero polynomial, then neither is their product. This leads us to our next result that is analogous to the rational numbers.
Theorem 8.19 Let \(F\) be a field. Then \(\langle F(x),+,\cdot \rangle\) is a field.
Proof. We leave the proof as an exercise to help the reader become more familiar with this field.
This implies that the rational expressions with real valued coefficients, \(\mathbb{R}(x)\) forms a field very similar to the rational numbers. We will explore this field in greater depth in Section 9.7.