1.4 Rational Expressions
1.4.1 Learning Objectives
By the end of this section, you should be able to:
- determine the domain of a rational expression;
- simplify rational expressions while retaining restrictions;
- multiply, divide, add, and subtract rational expressions;
- simplify compound fractions; and
- rationalize expressions involving binomial radicals.
1.4.2 Rational Expressions and Domain
A rational expression is a quotient of polynomials:
\[ \frac{P(x)}{Q(x)},\qquad Q(x)\ne0. \]
Its domain contains all real values for which the expression is defined. Values that make a denominator zero must be excluded.
Example 1.17 Domain
Find the domain of
\[ \frac{x+1}{x^2-5x+6}. \]
Solution
\[ x^2-5x+6=(x-2)(x-3), \]
so \(x\ne2,3\). The domain is
\[ \boxed{(-\infty,2)\cup(2,3)\cup(3,\infty)}. \]
Exercise 1.23 State the excluded value of \(\frac{4}{x-7}\).
Show answer
Answer: \(x\ne7\)
1.4.3 Simplifying Rational Expressions
To simplify a rational expression, factor its numerator and denominator and cancel common factors:
\[ \frac{AC}{BC}=\frac AB,\qquad B\ne0,\ C\ne0. \]
Only factors can be cancelled. Terms joined by addition or subtraction cannot be cancelled individually.
Example 1.18 Simplifying
\[ \frac{x^2-9}{x^2+x-6} =\frac{(x-3)(x+3)}{(x+3)(x-2)} =\boxed{\frac{x-3}{x-2}}, \]
with original restrictions \(x\ne-3,2\).
Exercise 1.24 Simplify \(\frac{x^2-16}{x^2+x-20}\).
Show answer
Answer: \((x+4)/(x+5)\) with original restrictions
1.4.4 Multiplying and Dividing Rational Expressions
Multiplication and division follow the ordinary fraction rules:
\[ \frac AB\cdot\frac CD=\frac{AC}{BD}, \qquad \frac AB\div\frac CD=\frac AB\cdot\frac DC. \]
Example 1.19 Multiplying
\[ \frac{x^2-4}{x^2-3x+2}\cdot\frac{x-1}{x+2} =\boxed{1}, \]
after factoring and cancelling common factors.
Exercise 1.25 Multiply \(\frac{3x}{x^2-9}\cdot\frac{x+3}{6}\).
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Answer: \(x/[2(x-3)]\)
Exercise 1.26 Divide \(\frac{x}{x-2}\div\frac{x+1}{x-2}\).
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Answer: \(x/(x+1)\)
1.4.5 Adding and Subtracting Rational Expressions
To add or subtract rational expressions, use a common denominator:
\[ \frac AC+\frac BC=\frac{A+B}{C}. \]
For different denominators, factor first and build the least common denominator from the greatest required power of each factor.
Example 1.20 Adding
\[ \frac2{x-1}+\frac3{x+2} =\frac{2(x+2)+3(x-1)}{(x-1)(x+2)} =\boxed{\frac{5x+1}{(x-1)(x+2)}}. \]
Exercise 1.27 Add \(\frac1x+\frac2{x+1}\).
Show answer
Answer: \((3x+1)/[x(x+1)]\)
1.4.6 Compound Fractions and Rationalizing Binomials
A compound fraction contains fractions in its numerator, denominator, or both. It can be simplified by combining its smaller fractions or by multiplying its numerator and denominator by the LCD.
Example 1.21 Compound Fraction
\[ \frac{\frac1x+\frac1y}{\frac1x-\frac1y} =\frac{y+x}{y-x}, \]
found by multiplying the numerator and denominator by \(xy\).
To rationalize a binomial involving square roots, multiply by its conjugate:
\[ (A+B)(A-B)=A^2-B^2. \]
1.4.7 Conceptual Takeaways
- Domain restrictions come from the original expression.
- Factoring reveals common factors and least common denominators.
- Cancelling removes common factors, not separate terms.
- Division by a rational expression means multiplying by its reciprocal.
- Addition and subtraction require a common denominator.
- Conjugates remove square roots from binomial products.
1.4.8 Skills You Should Be Able to Do
- Find the domain of an algebraic expression.
- Simplify rational expressions while retaining restrictions.
- Multiply, divide, add, and subtract rational expressions.
- Find an LCD from factored denominators.
- Simplify compound fractions.
- Rationalize a numerator or denominator using a conjugate.
1.4.9 Practice Problems with Solutions
Find the domain of \(\frac{2x}{x^2-x-12}\).
Show Solution
\[ x^2-x-12=(x-4)(x+3), \] so \(\boxed{x\ne4,-3}\).
Simplify \(\frac{x^2+5x+6}{x^2+4x+3}\).
Show Solution
\[ \frac{(x+2)(x+3)}{(x+1)(x+3)} =\boxed{\frac{x+2}{x+1}}, \] with \(x\ne-3,-1\).
Multiply \(\frac{x^2-1}{x^2+3x+2}\cdot\frac{x+2}{x-1}\).
Show Solution
\[ \frac{(x-1)(x+1)}{(x+1)(x+2)} \cdot\frac{x+2}{x-1} =\boxed{1}, \] for values in the original domain.
Divide \(\frac{2x}{x^2-9}\div\frac{4}{x+3}\).
Show Solution
\[ \frac{2x}{(x-3)(x+3)}\cdot\frac{x+3}{4} =\boxed{\frac{x}{2(x-3)}}. \]
Add \(\frac3x+\frac2{x-4}\).
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\[ \frac{3(x-4)+2x}{x(x-4)} =\boxed{\frac{5x-12}{x(x-4)}}. \]
Subtract \(\frac{x}{x+1}-\frac1{x-1}\).
Show Solution
\[ \frac{x(x-1)-(x+1)}{(x+1)(x-1)} =\boxed{\frac{x^2-2x-1}{x^2-1}}. \]
Simplify \(\frac{\frac2x+\frac1y}{\frac1x}\).
Show Solution
Multiply the numerator and denominator by \(xy\): \[ \frac{2y+x}{y}=\boxed{2+\frac{x}{y}}. \]
Rationalize \(\frac3{\sqrt5-1}\).
Show Solution
\[ \frac3{\sqrt5-1}\cdot\frac{\sqrt5+1}{\sqrt5+1} =\boxed{\frac{3(\sqrt5+1)}4}. \]