Working With Rational Expressions.pdf - ADVFUN12U

Working with Rational Expressions

A rational expression is an algebraic expression that can be written as the quotient of two polynomials. A rational expression is undefined if the denominator is zero, so we write restrictions on the variables to avoid this. Simplifying Rational Expressions A rational expression can be simplified by factoring the numerator and the denominator, and then dividing out the common factors. Example 1: Simplify and state restrictions. A)

B)

C)

Solution: A)

B)

C)

-1

Factor the numerator and the denominator to find the largest possible common factor to divide out. Write restrictions on the variables to prevent the denominator from equalling zero.

Opposites When factors are opposites, you can factor out (-1) from one of the factors to make the factors identical.

1

Mr. White | MHF4U | Unit 0 – Basics | Rational Expressions

Multiplying and Dividing Rational Expressions To multiply or divide rational expressions, factor the numerators and the denominators (where possible), and then look for common factors that can be divided out. Example 2: Simplify Solution Factor each polynomial

1

1 1

1

Divide out the common factors to reduce the expression to lowest terms.

2

Write restrictions to prevent the denominator from equalling zero, which would result in undefined values.

1

Example 3: Simplify Solution Change the division into multiplication by the reciprocal. Factor.

1

Divide out the identical factors.

1

1

1 Write restrictions to avoid undefined values.

2

Mr. White | MHF4U | Unit 0 – Basics | Rational Expressions

Adding and Subtracting Rational Expressions To add or subtract rational expressions, you must have a common denominator. To ensure that you will use the lowest common denominator, factor the numerators and the denominators first. This will keep the expressions as simple as possible. Example 4: Simplify Solution Factor the numerators and denominators.

1 1

If possible, divide out like terms (but only within each rational expression) The lowest common denominator is 7(x-2). Multiply the numerator and denominator of each rational expression to create an equivalent expression with the desired common denominator. Simplify the numerator Write restrictions to avoid undefined values.

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Mr. White | MHF4U | Unit 0 – Basics | Rational Expressions

Practise: 1. State the restrictions (if any) on each expression a)

b)

c)

d)

2. Simplify, and state restrictions. Write your answers using positive exponents. a)

b)

c)

d)

e)

f)

3. Simplify, and state restrictions. a) b) c) d) e) f) 4. Show that

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Mr. White | MHF4U | Unit 0 – Basics | Rational Expressions