# Online Lecture Chapter R Algebraic Expressions

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### Online Lecture Chapter R Algebraic Expressions

• 1. Chapter R Review: Algebraic Expressions • Goals – Factor a polynomial – Factor and simplify rational expressions – Simplify complex fractions – Rationalize an expression involving radicals – Simplify an algebraic expression with negative exponents
• 2. Factoring Polynomials The first few problems in your Chapter R homework assignment have to do with factoring. Please take advantage of the built-in help in MyMathLab (Help Me Solve This, View an Example, the link to the eBook) if you need assistance with these problems. Also, feel free to contact me if you need additional assistance. This is a critical skill in this course and in Calculus.
• 3. Rational Expressions • A rational expression is a quotient, , of two polynomials and . • When simplifying a rational expression, begin by factoring out the greatest common factor in the numerator and denominator. • Next, factor the numerator and denominator completely. Then cancel any common factors. ( ) ( ) p x q x ( )p x ( )q x
• 4. Example • Simplify the expression: − + 2 25 100 5 10 t t ( ) ( )22 Solution: 25 4 25 2 ( 2)25 100 5 10 5( 2) 5( 2) 5 25 t t tt t t t − − +− = = + + + = ( )2 ( 2)t t− + 5 ( 2)t + 5( 2)t= − (Solution will appear when you click)
• 5. Another Example • Simplify the expression: − − 3 4 12 x x Solution: 3 3 4 12 4( 3) Note that ( ) . Thus, 3 (3 ). 3 3 So we have 4( 3) x x x x a b b a x x x x x − − = − − − = − − − = − − − − = − 4 (3 )x× − − 1 . 4 = −    
• 6. Domain and Examples • The domain of a rational expression consists of all real numbers except those that make the denominator zero… – …since division by zero is never allowed! • Examples: ( ) ( ) p x q x
• 7. Video Examples: Factoring Please view the video linked below for an overview of factoring, leading up to a more complicated, Precalculus-level, factoring problem. http://create.lensoo.com/watch/b55K
• 8. Simplifying a Complex Fraction 5 2 Example: Simplify . 1 4 3 x x − + 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 5 x 1 3x 3 .x 3 .x
• 9. Solution, continued 5 52 23 3 3 3 11 44 33 33 x x xx x x x x xx   − × × − × ÷   =   × + ×+ × ÷   5 x = 3 x× 2 3 1 3 x x − × 3x× 15 6 1 124 3 x xx − = ++ ×
• 10. Expressions with Negative Exponents • Recall that • If the expression contains negative exponents, one way to simplify is to rewrite the expression as a complex fraction. • Recall the following properties of exponents: − = 1n n x x + − ⋅ = =and a a b a b a b b x x x x x x
• 11. Example: Simplify − − − − + 2 5 3 2 x y x y The previous page in Blackboard contained the following YouTube video with a solution to the example problem above: https://youtu.be/4O-2V1G18Qc
• 12. Rationalizing the Denominator • Given a fraction whose denominator is of the form or we sometimes want to rewrite the fraction with no square roots in the denominator. • This is called rationalizing the denominator of the given fractional expression. • It often allows the fraction to be simplified. a b+ +a b
• 13. Rationalizing (cont’d) • To rationalize the denominator, multiply both numerator and denominator of the fraction by the conjugate of the denominator. • The conjugate of is the conjugate of is a b+ ;a b− + .a b−a b
• 14. Example • Rationalize the denominator of • Note that we can also rationalize numerators in the same way! + 1 x y
• 15. Another Example • Rationalize the denominator of − + 2 2 b c b c ( )( ) ( )( ) ( ) 2 2 2 2 Multiply numerator and denominator by the conjugate of the denominator Factor the difference of squar . . Multiply out the d es enominator. b c b c b c b c b c b c b b c b c c − − × + − − + − = − − = ( )( ) bc b b c c + − − ( )( )b c b c= + −
• 16. Example Write the expression as a single quotient in which only positive exponents and/or radicals appear: Assume that Video solution: https://youtu.be/ecDpNBG3R-U (contained on previous slide in Blackboard) ( ) ( ) − + − + + 1/2 1/2 5 7 5 5 x x x x 5.x > −
• 17. First, notice that the numerator has a common factor of x + 5. Always take the lower exponent when taking out a common factor. In this case, the exponents are 1/2 and -1/2, so the lower exponent is -1/2. ( ) ( ) − + − + + 1/2 1/2 5 7 5 5 x x x x Written solution to previous example:
• 18. 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. Remember, we are factoring out continued on next slide. . . ( ) ( ) ( ) − − − − −−  + + − +   + 1 1/2 1/2/2 ( 1/2) ( 1/2) 5 5 7 5 5 x x x x x ( ) 1/2 5 .x − +
• 19. ( ) ( ) ( ) ( ) ( ) ( ) − − − − −− −  + + − +   +  + + − +  = + 1/2 ( 1/2) ( 1/21/2 1/2 1/2 1 0 ) 5 5 7 5 5 5 5 7 5 5 x x x x x x x x x x ( ) ( ) + − = + + 1/2 5 7 5 5 x x x x − = + 3/2 5 6 ( 5) x x
• 20. Example Write the expression as a single quotient in which only positive exponents and/or radicals appear: Assume that − +1/2 1/24 2 3 x x 0.x >
• 21. Again, factor out the common factor with the lower exponent: Now combine the expressions to form a single quotient. − − − − −− −  + = +    1/2 (1/2 1/2 1/2 11 //2) ( 1/2 2)4 4 2 2 3 3 x x x x x 1/2 x− − −    = + = +        1/2 0 1 1/24 4 2 2 3 3 x x x x x   = +    1 4 2 3 x x
• 22. ⋅    + = +   ⋅    1 4 1 2 3 4 2 3 1 3 3 x x x x + +  = =    1 6 4 6 4 3 3 x x x x
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