Chapter 1
Linear Equations and Graphing
Course: Gre
Subject: Gre Quantitative Reasoning
Chapter: Rational Expressions And Equations
Introduction: Extending Fraction Rules
Multiplying and dividing rational expressions follow the same rules as multiplying and dividing numerical fractions. However, because the numerators and denominators are polynomials, factoring becomes an essential step.
Multiplication of Rational Expressions
To multiply rational expressions:
- Factor all numerators and denominators completely
- Cancel common factors (from numerator to denominator only)
- Multiply the remaining factors (straight across)
PQ×RS=P×RQ×S
Division of Rational Expressions
To divide rational expressions:
- Invert (take the reciprocal of) the second fraction
- Multiply as above
PQ÷RS=PQ×SR
Example 1: Multiplication
Simplify: y-1y2-7y+10×y2-8y+15y2-4y+3
Step 1: Factor
y2-7y+10=(y-5)(y-2) y2-8y+15=(y-3)(y-5) y2-4y+3=(y-1)(y-3)
Expression becomes:
y-1y-5)(y-2×y-3)(y-5y-1)(y-3
Step 2: Cancel common factors
Cancel y1, y5, y3
Step 3: Multiply
1y-2
Example 2: Division
Simplify: r-4r2-5r+6÷r-3r2-6r+9
Step 1: Factor
r2-5r+6=(r-3)(r-2) r2-6r+9=(r-3)(r-3)
Step 2: Invert second fraction and change to multiplication