Unit 11.1 Introduction to Rational Expressions
Rational Expressions
Rational expressions are fractions in which the numerator and the denominator are both polynomials. Understanding rational expressions is crucial for solving various algebraic problems and real-world applications.
A rational expression is an expression of the form
[latex]\frac{f(x)}{g(x)}[/latex]
where f(x) and g(x) are both polynomials and g(x) ≠ 0. This is what we call restriction. Any values that cause the denominator to be zero will be a non-solution
Simplifying Rational Expressions
To simplify a rational expression, follow these steps:
- Factor the Numerator and Denominator:
- Factor both the numerator and the denominator completely.
- Cancel Common Factors:
- Identify and cancel any common factors between the numerator and the denominator.
- State the Restrictions:
- Determine the values of the variable that would make the denominator zero, in the original equation, and exclude these values from the domain.
Simplifying a rational expression
Simplify the expression [latex]\frac{x^2 - 9}{x^2 - 3x}[/latex]
Factor the numerator and denominator
Numerator
[latex]x^2 - 9 = (x + 3)(x - 3)[/latex]
Denominator
[latex]x^2 - 3x = x(x - 3)[/latex]
Rewrite and cancel out the common factors
[latex]\frac{(x + 3)(x - 3)}{x(x - 3)}[/latex]
[latex]\frac{(x + 3)\cancel{(x - 3)}}{x\cancel{(x - 3)}}[/latex]
[latex]\frac{x + 3}{x}[/latex]
Since the x on the top is part of a polynomial, we cannot simplify it further.
State the restrictions
x ≠ 0, 3