Rational expressions are fractions with polynomials in the numerator and denominator. They're key for modeling complex relationships in math and science. Simplifying and operating on these expressions involves factoring, canceling common terms, and finding common denominators.
Adding, subtracting, multiplying, and dividing rational expressions requires specific techniques. Solving equations with rational expressions often involves clearing fractions by multiplying both sides by the . It's crucial to check solutions and identify restrictions on variables.
Simplifying and Operating on Rational Expressions
Simplification of rational expressions
Factor numerator and denominator completely
Identify greatest common factor (GCF) of terms in numerator and denominator (x2+2x, GCF is x)
Factor out GCF from numerator and denominator (x(x+2))
Factor remaining expressions in numerator and denominator (x2−4=(x+2)(x−2))
Cancel common factors in numerator and denominator
Identify matching factors in numerator and denominator ((x+2) in both)
Divide out (cancel) matching factors, leaving remaining factors in numerator and denominator (x−2x)
Combine remaining factors in numerator and denominator
Multiply remaining factors in numerator to form new numerator (x)
Multiply remaining factors in denominator to form new denominator (x−2)
Operations with rational expressions
Multiplication of rational expressions
Factor numerators and denominators of rational expressions (x−3x+1⋅x+4x−2)
Multiply numerators together to form new numerator ((x+1)(x−2))
Multiply denominators together to form new denominator ((x−3)(x+4))
Simplify resulting by canceling common factors (x2+x−12x2−x−2)
Division of rational expressions
Rewrite division as multiplication by reciprocal of divisor (x−1x+2÷x−4x+3=x−1x+2⋅x+3x−4)
Reciprocal: flip numerator and denominator of divisor (x−4x+3→x+3x−4)
Multiply rational expressions using multiplication rules (see above)
Simplify resulting rational expression by canceling common factors ((x−1)(x+3)(x+2)(x−4))
Addition and subtraction of rationals
Like denominators
Add or subtract numerators while keeping denominator the same (x2+x3=x2+3=x5)
Simplify resulting numerator and denominator if possible
Unlike denominators
Find least common denominator (LCD) of rational expressions
LCD is the least common multiple (LCM) of individual denominators (x−11−x+12, LCD is (x−1)(x+1))
Rewrite each rational expression as an equivalent expression with LCD
Multiply numerator and denominator by factor needed to create LCD (x−11⋅x+1x+1−x+12⋅x−1x−1)
Add or subtract numerators of equivalent expressions while keeping LCD as denominator ((x−1)(x+1)x+1−(x−1)(x+1)2(x−1)=(x−1)(x+1)x+1−2(x−1))
Simplify resulting numerator and denominator if possible ((x−1)(x+1)−x+3)
Complex rational expressions
Simplify numerator and denominator separately using rational expression simplification rules
Factor and cancel common factors in numerator and denominator (x−4x+3x−1x+2, simplify to x+3x+2⋅x−1x−4)
Perform division of simplified rational expressions
Rewrite as multiplication by reciprocal and simplify ((x+3)(x−1)(x+2)(x−4))
Understanding Rational Expressions
A rational expression is a fraction (ratio) of two polynomials
The numerator and denominator are algebraic expressions containing variables
Rational expressions are a type of algebraic expression used to represent complex relationships
Equations involving rational expressions can model real-world scenarios and require specific solving techniques
Evaluating and Solving Rational Expressions
Equations with rational expressions
Clear fractions by multiplying both sides of the equation by the LCD of all rational expressions
Identify LCD of all rational expressions in the equation (x−21+x+13=x−22, LCD is (x−2)(x+1))
Multiply both sides of the equation by the LCD ((x−2)(x+1)⋅(x−21+x+13)=(x−2)(x+1)⋅x−22)
Simplify each side of the equation by distributing and combining like terms (x+1+3(x−2)=2(x+1))
Solve resulting equation using appropriate method (e.g., factoring, quadratic formula)
Subtract 2x and 2 from both sides (x+1+3x−6=2x+2→4x−5=2x+2)
Subtract 2x from both sides (2x−5=2)
Add 5 to both sides (2x=7)
Divide both sides by 2 (x=27)
Check potential solutions by substituting them back into the original equation
Verify that each solution satisfies the equation (27−21+27+13=27−22→231+293=232→32+32=34, which is true)
Identify restrictions on variables that would make any denominator equal to zero
These values are not part of the solution set of the equation (x=2 and x=−1 in the original equation)
Key Terms to Review (2)
Least common denominator: The least common denominator (LCD) is the smallest positive number that is a multiple of the denominators of two or more fractions. It is used to add, subtract, or compare fractions.
Rational expression: A rational expression is a fraction where both the numerator and the denominator are polynomials. It is defined for all values of the variable except those that make the denominator zero.