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# A.3 rational expressions

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Simplifying algebraic expressions.
This is a study guide for section A.3.

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### A.3 rational expressions

1. 1. Section A.3 Rational Expressions <ul><li>Goals </li></ul><ul><ul><li>Factor and simplify rational expressions </li></ul></ul><ul><ul><li>Simplify complex fractions </li></ul></ul><ul><ul><li>Rationalize an expression involving radicals </li></ul></ul><ul><ul><li>Simplify an algebraic expression with negative exponents </li></ul></ul>
2. 2. Rational Expressions <ul><li>A rational expression is a quotient p / q of two polynomials p and q . </li></ul><ul><li>When simplifying a rational expression, begin by factoring out the greatest common factor in the numerator and denominator. </li></ul><ul><li>Next, factor the numerator and denominator completely. Then cancel any common factors. </li></ul>
3. 3. Domain and Examples <ul><li>The domain of a rational expression p / q consists of all real numbers except those that make the denominator zero… </li></ul><ul><ul><li>… since division by zero is never allowed! </li></ul></ul><ul><li>Examples: </li></ul>
4. 4. Example <ul><li>Simplify the expression: </li></ul>(Solution will appear when you click)
5. 5. Another Example <ul><li>Simplify the expression: </li></ul>
6. 6. Simplifying a Complex Fraction To simplify a complex fraction, multiply the entire numerator and denominator by the least common denominator of the inner fractions. The inner fractions are and . Their LCD is Thus, we should multiply the entire numerator and denominator by
7. 8. Expressions with Negative Exponents <ul><li>Recall that </li></ul><ul><li>If the expression contains negative exponents, one way to simplify is to rewrite the expression as a complex fraction. </li></ul><ul><li>Recall the following properties of exponents: </li></ul>
8. 9. Example: Simplify First, rewrite the expression as a complex fraction. Multiply numerator and denominator by the LCD. The least common denominator of the inner fractions is
9. 10. Rationalizing the Denominator <ul><li>Given a fraction whose denominator is of the form or we sometimes want to rewrite the fraction with no square roots in the denominator. </li></ul><ul><li>This is called rationalizing the denominator of the given fractional expression. </li></ul><ul><li>It often allows the fraction to be simplified. </li></ul>
10. 11. Rationalizing (cont’d) <ul><li>To rationalize the denominator, multiply both numerator and denominator of the fraction by the conjugate of the denominator. </li></ul><ul><li>The conjugate of is the conjugate of is </li></ul>
11. 12. Example <ul><li>Rationalize the denominator of </li></ul><ul><li>Note that we can also rationalize numerators in the same way! </li></ul>
12. 13. Another Example <ul><li>Rationalize the denominator of </li></ul>
13. 14. Example <ul><li>Write the expression as a single quotient in which only positive exponents and/or radicals appear: </li></ul><ul><li>Assume that </li></ul>
14. 15. <ul><li>First, notice that the numerator has a common factor of x + 5. </li></ul><ul><li>Always take the lower exponent when taking out a common factor. </li></ul><ul><li>In this case, the exponents are 1/2 and -1/2, so the lower exponent is -1/2. </li></ul>
15. 16. <ul><li>To find the exponent that is left when you take out a common factor to a certain power, subtract that power from each exponent of the common factor. </li></ul><ul><li>Remember, we are factoring out </li></ul><ul><li>continued on next slide. . . </li></ul>
16. 18. Example <ul><li>Write the expression as a single quotient in which only positive exponents and/or radicals appear: </li></ul><ul><li>Assume that </li></ul>
17. 19. <ul><li>Again, factor out the common factor with the lower exponent : </li></ul><ul><li>Now combine the expressions to form a single quotient . </li></ul>