Rational expressions are fractions that contain variables instead of just integers in the numerator and denominator. Just like fractions, rational expressions are a result of dividing the numerator by the denominator. Operations such as addition, subtraction, and multiplication can be performed on rational expressions as well. You can also use tools that apply to other equations, such as commutative, distributive, and associative properties. Equations involving rational expressions can be simplified by finding the least common denominators, making them easier to solve. It is important to follow the order of operations, even though it is a fraction.
Rational Expression: An expression that is the ratio of two polynomials and the denominator is not \(0\).
Excluded Value: A value that makes a rational expression undefined (\(denominator = 0\)).
Rational Equation: An equation with one or more rational expressions.
Rational expressions are basically the quotient (result of division) of two polynomials.
When we simplify rational expressions, we want the numerator and denominator to not have any factors in common.
Excluded values are the values that make the denominator zero. We cannot divide by zero, so we exclude any valuesthat cause the denominator to become zero.
When we cancel common factors, we may remove a division by zero.
Example: \(\frac{4x - 2}{2x^2 + x -1}\)
The denominator equals \(0\) when \(x = \frac{1}{2}\) or \(x = -1\)
\(\frac{4x - 2}{2x^2 + x - 1} = \frac{2(2x - 1)}{(2x - 1)(x + 1)}\)
If we cancel common factors first, we get \(\frac{2}{x + 1}\)
Now the denominator equals \(0\) when \(x = 1\). We have removed \(x = \frac{1}{2}\) as an excluded value.
Technically the original expression and the simplified one are not the same. We should write:
\(\frac{4x - 2}{2x^2 + x - 1} = \frac{2}{x + 1}, x \neq \frac{1}{2}\)
Let \(a, b, c,\) and \(d\) be polynomials.
Sums and differences with the same denominator:
Sums and differences with different denominators:
To find the LCD, we find the least common multiple of the denominators of the different fractions. The least common multiple of two or more integers is the least positive integer that has all of those integers as factors.
Example: \( \frac{2}{x + 2} - \frac{3}{2x - 5}\)
The denominators cannot be factored, so the LCD is the product of the denominators: \((x+2)(2x-5)\).
\( \frac{2}{x + 2} . \frac{(2x - 5)}{(2x - 5)} - \frac{3}{2x - 5} . \frac{(x + 2)}{(x + 2)}\)
\( \frac{2(2x - 5) - 3(x + 2)}{(x + 2)(2x - 5)} = \frac{4x - 10 - 3x - 6}{(x + 2)(2x - 5)} = \frac{x - 16}{(x + 2)(2x - 5)}\)
Rational equations with one term on each side:
Example: \(\frac{x}{5}\)
\(=\)
\(\frac{x + 1}{2}\)
\(\frac{x}{5} . 5\)
\(=\)
\(\frac{x + 1}{2} . 5\)
\(x . 5\)
\(=\)
\(\frac{5(x + 1)}{2} . 2\)
\(2x\)
\(=\)
\(5(x + 1)\)
\(2x\)
\(=\)
\(5x + 5\)
\(-5\)
\(=\)
\(3x\)
\(x\)
\(=\)
\(- \frac{3}{5}\)
We can use lowest common denominators to solve rational equations.
Example: \(\frac{x}{x - 2} + \frac{1}{5} = \frac{2}{x - 2}\)
The LCD is 5(x-2), so multiply the original equation by the LCD.
\(\frac{x}{x - 2} . 5(x - 2) + \frac{1}{5} . 5(x - 2)\)
\(=\)
\(\frac{2}{x - 2} . 5(x - 2)\)
\(5x + (x - 2)\)
\(=\)
\(2 . 5\)
\(6x - 2\)
\(=\)
\(10\)
\(x\)
\(=\)
\(\frac{12}{6} = 2\)
Be careful of extraneous solutions! \(x = 2\) looks like a solution, but this would make \(\frac{x}{x - 2}\) and \(\frac{2}{x - 2}\) undefined.
Since \(x = 2\) is actually an extraneous solution, this rational equation has no solution!