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Introduction to Rational Expressions
A rational expression can be written as P (x)/Q (x), where P and Q are polynomials and Q (x) is not allowed to take on the value zero. e.g. 5x+7 , 3x-8 , 6x - 8 x x3+27
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Table of Contents Click to jump to a specific topic or press enter to go through in order
Restrictions Simplifying Multiplying Dividing Adding/Subtracting Solving Rational Equations previous next
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Restricted Value Restricted Value is a value that is not a permissible replacement for a variable. Table of contents previous next
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How to Determine Restricted Values
1. Set the denominator equal to zero. 2. Solve the resulting equation. 3. Any solution of the equation is a restricted value. Table of contents previous next
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Working with Rational Expressions:
Be able to: Simplify rational expressions Multiply rational expressions Divide rational expressions Table of contents previous next
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How to Simplify Rational Expressions
1. Factor the numerator and the denominator completely. 2. Divide out any factor that is common to both. Table of contents previous next
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Simplify the Rational Expression
= = Table of contents previous next
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Simplify the Rational Expression
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Simplify the Rational Expression
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Definition of Multiplication of Rational Expressions
If a,b,c, and d represent algebraic expressions, where b and d are not 0, then Table of contents previous next
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How to Multiply Rational Expressions
1. Factor each numerator and denominator completely. 2. Divide out any factors that are common to both the numerator and denominator. 3. Combine the numerators and denominators. Table of contents previous next
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Multiply and Simplify 5 Table of contents previous next
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Multiply and Simplify -1 -1 Table of contents previous next
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Definition of Division of Rational Expressions
If a,b,c, and d represent algebraic expressions, where b,c, and d are not 0, then: Multiply by the reciprocal of the divisor Table of contents previous next
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Divide and Simplify Table of contents previous next
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Divide and Simplify Table of contents previous next
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Addition & Subtraction of Rational Expressions
Be able to: Add & subtract rational expressions with like denominators Determine the Lowest Common Multiple Write equivalent rational expressions using the LCM as the new denominator Add & subtract rational expressions with unlike denominators Table of contents previous next
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How to Add or Subtract Rational Expressions with Like Denominators
1. Add(or subtract) the numerators. 2. Keep the common denominator. 3. Simplify the result. Table of contents previous next
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Perform the Indicated Operations
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Perform the Indicated Operations
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Perform the Indicated Operation
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Add or Subtract with Unlike Denominators
1. Determine the Least Common Denominator (LCD). 2. Rewrite each expression with the LCD as its denominator. 3. Expand and simplify numerators. Factor if you can 4. Simplify by canceling. Table of contents previous next
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Determining the LCM of Polynomials
1. Factor 2. Include in the LCM each factor that appears in at least one polynomial. 3. For each factor, use the largest exponent that appears. Table of contents previous next
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Find the LCM x-7, 4x-28, 6x x-7 = (x-7) 4x-28 = 2.2.(x-7) 6x = 2.3.x
LCM is: x.(x-7) 12x(x-7) Table of contents previous next
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Find the LCM Table of contents previous next
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Find the LCM List all factors seen with the most number of times that factor is seen in any one expression. Table of contents previous next
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1. Perform the Indicated Operation
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1. Perform the Indicated Operation
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1. Perform the Indicated Operation
= = Table of contents previous next
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1. Perform the Indicated Operation
= = Table of contents previous next
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2. Perform the Indicated Operation
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3. Perform the Indicated Operation
= = Table of contents previous next
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3. Perform the Indicated Operation
= = Table of contents previous next
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4. Perform the Indicated Operation
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4. Perform the Indicated Operation
= = = Table of contents previous next
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Problem The width of a rectangle is given by metres and its length is given by metres. Find an expression for its perimeter. Table of contents previous next
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Problem (continued) Table of contents previous next
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Problem (continued) Table of contents previous next
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Problem (continued) Table of contents previous next
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Solving a Rational Equation Algebraically
Determine the restricted values. Find LCM of denominators. Multiply both sides of the equation by LCM. Cancel denominators. Solve the resulting equation. Discard solutions that are restricted values. Check Table of contents previous next
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1. Solve Algebraically Step 1: Find restricted values
Step 2: Find the LCM of (x + 1), x, and x(x + 1) LCM = x(x + 1) Step 3: Multiply both sides of equation by the LCM Table of contents previous next
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All the denominators cancel out!
1. Solve Algebraically All the denominators cancel out! Table of contents previous next
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1. Solve Algebraically Remember to check against restricted values
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2. Solve Algebraically Step 1: Find the restricted values
Step 2: Find the LCM Step 3: Multiply LCM by both sides of equation Table of contents previous next
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All the denominators cancel out!
2. Solve Algebraically All the denominators cancel out! Table of contents previous next
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2. Solve Algebraically Remember to check against restricted values
Since x = 2 is a restriction, it is referred to as an extraneous solution No solution Table of contents previous next
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3. Solve Algebraically Step 1: Find the restricted values
Step 2: Find the LCM Step 3: Multiply LCM by both sides of equation Table of contents previous next
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3. Solve Algebraically Table of contents previous next
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3. Solve Algebraically Equation is true for all Real numbers.
Are you sure? What about restricted values? All real numbers except b = 3,-3. Table of contents previous next
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4. Solve Table of contents previous next
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4. Solve (continued) Table of contents previous next
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Problem 1 Paul can wax his car in 45 minutes. His big brother John can do the job in 30 minutes. If they work together, how long will it take them to wax Paul’s car? Table of contents previous next
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Problem 1 x = time to wax the car working together(minutes)
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Problem 1 Solving algebraically: Table of contents previous next
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Problem 1 It will take Paul & John 18 minutes to wax Paul’s car.
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Problem 2 It takes one person twice as long to shovel snow from the driveway as it takes another using a snow blower. If the two of them together can clear the driveway in 8 minutes, how long does it take the person shoveling alone? Table of contents previous next
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Problem 2 x = time for person using the snow
blower to clear the driveway (mins) 2x = time for person shoveling to complete the driveway (mins) Table of contents previous next
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Problem 2 Table of contents previous next
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Problem 2 The person shoveling alone will take 24 minutes the shovel the driveway. Table of contents previous
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