Rational expressions are like fractions, but with algebraic terms instead of numbers. They're a key part of algebra, helping us solve complex problems and model real-world situations.
Simplifying these expressions is crucial for easier calculations and clearer understanding. We'll learn how to factor, cancel out common terms, and handle tricky situations like values and complex fractions.
Simplifying Rational Expressions
Introduction to Rational Expressions
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A is an where both the and are polynomials
of rational expressions involves reducing them to their simplest form
Understanding variable restrictions is crucial when working with rational expressions
Undefined values in rational expressions
Rational expressions become undefined when the denominator equals zero
Set the denominator equal to zero and solve for the variable to find the undefined values
x−52x+1 is undefined when x−5=0, so x=5 is the undefined value
Evaluation of rational expressions
Substitute the given value for the variable and simplify to evaluate a rational expression
Expression is undefined if the substituted value results in the denominator being zero
Factor the numerator and denominator completely to simplify a rational expression
Cancel out common factors in the numerator and denominator
Simplified expression is formed by the remaining factors in the numerator and denominator
Simplify x+3x2−9
Factor the numerator: x+3(x+3)(x−3)
Cancel the (x+3): x−3
Simplification with opposite factors
Opposite factors are binomials with the same terms but opposite signs (x+2) and (x−2)
Rational expressions containing opposite factors can be canceled out
Simplified expression will have a negative sign in the numerator or denominator
Simplify x+4x−4
Numerator and denominator are opposite factors
Cancel them out, resulting in −1
Techniques for complex rational expressions
Complex rational expressions contain rational expressions in the numerator, denominator, or both
Simplify complex rational expressions by:
the numerator and denominator completely
Identifying and canceling out common factors
Combining the remaining factors to form the simplified expression
Simplify x+1x−1x+3x−3
Multiply the numerator and denominator by the LCD of (x+3)(x+1)
(x−1)(x+3)(x−3)(x+1)
Factor the numerator and denominator: (x+3)(x−1)(x+1)(x−3)