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Rational Expressions

Learn how to simplify rational expressions by dividing out common factors, factoring numerator and denominator, avoiding undefined values, and understanding exclusions.

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Rational Expressions

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  1. Rational Expressions Simplifying Joyce DuVall Green Valley High School Henderson, Nevada

  2. Simplifying Rational Expressions • The objective is to be able to simplify a rational expression

  3. Vocabulary • Polynomial – The sum or difference of monomials. • Rational expression – A fraction whose numerator and denominator are polynomials. • Domain of a rational expression – the set of all real numbers except those for which the denominator is zero. • Reduced form – a rational expression in which the numerator and denominator have no factors in common.

  4. Simplifying Rational Expressions • Divide out the common factors • Factor the numerator and denominator and then divide the common factors

  5. Dividing Out Common Factors Step 1 – Identify any factors which are common to both the numerator and the denominator. • The numerator and denominator have a common factor. • The common factor is the five.

  6. Dividing Out Common Factors • Step 2 – Divide out the common factors. • The fives can be divided since 5/5 = 1 • The x remains in the numerator. • The (x-7) remains in the denominator

  7. Factoring the Numerator and Denominator • Factor the numerator. • Factor the denominator. • Divide out the common factors. • Write in simplified form.

  8. Factoring Step 1: Look for common factors to both terms in the numerator. • 3 is a factor of both 3 and 9. • X is a factor of both x2 and x. Step 2: Factor the numerator. + 3 x ( x 3 ) 3 12 x

  9. Factoring Step 3: Look for common factors to the terms in the denominator and factor. • The denominator only has one term. The 12 and x3 can be factored. • The 12 can be factored into 3 x 4. • The x3 can be written as x • x2. + 3 x ( x 3 ) 2 · · · 3 4 x x

  10. Divide and Simplify Step 4: Divide out the common factors. In this case, the common factors divide to become 1. Step 5: Write in simplified form. + x 3 2 4 x

  11. You Try It Simplify the following rational expressions.

  12. For what value of x is undefined? Restrictions on Rational Expressions It is undefined for any value of “x” which makes the denominator zero. The restriction is that x cannot equal 5.

  13. YOU TRY IT What are the excluded values of the variables for the following rational expressions?

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