L6 Rational Expressions - Part 3
Completion requirements
Unit 1A
Precalculus
Lesson 6: Rational Expressions
Multiplying and Dividing Rational Expressions
MultiplyingJust as when multiplying fractions, multiplying rational expressions employs the same rule of “top times top, bottom times bottom”. When multiplying rational expressions, factor the numerator and the denominator, identify the non-permissible values, eliminate factors found in both the numerator and denominator, and then simplify.
Dividing
Similarly, dividing rational expressions employs the same rule as dividing fractions. That is, multiply the first expression by the reciprocal of the second expression. However, one noticeable difference is that non-permissible values must be determined for any factor that appears in any denominator at any point in the process. Example 5 will illustrate the identification of non-permissible values of rational expressions undergoing division.
Simplify the rational expression «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mrow»«mrow»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»18«/mn»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math».
Identify any non-permissible values.
Step 1:
Factor both the numerator and the denominator.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mrow»«mrow»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»18«/mn»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mrow»«mrow»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»4«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mn»6«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mn»3«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mn»2«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»
Step 2:
Identify the NPVs.
The NPVs are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left center center right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«mtd/»«mtd/»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»3«/mn»«/mtd»«mtd/»«mtd/»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The NPVs are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
Step 3:
Simplify by eliminating factors found in both the numerator and denominator.
Because the numerators are multiplied and the denominators are multiplied, a factor in the numerator of the first expression can be used to eliminate a factor in the denominator of the second expression, as is shown by the elimination of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» below.
The simplified expression is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
Because the numerators are multiplied and the denominators are multiplied, a factor in the numerator of the first expression can be used to eliminate a factor in the denominator of the second expression, as is shown by the elimination of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» below.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mn»4«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mn»6«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mn»3«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mn»2«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mmultiscripts»«menclose notation=¨updiagonalstrike¨»«mn»4«/mn»«/menclose»«mprescripts/»«none/»«mn mathcolor=¨#FF0000¨»2«/mn»«/mmultiscripts»«menclose notation=¨updiagonalstrike¨»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/menclose»«/mrow»«mrow»«mmultiscripts»«menclose notation=¨updiagonalstrike¨»«mn»6«/mn»«/menclose»«mprescripts/»«none/»«mn mathcolor=¨#FF0000¨»2«/mn»«/mmultiscripts»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«mo»§#8729;«/mo»«mfrac»«mrow»«mmultiscripts»«menclose notation=¨updiagonalstrike¨»«mn»3«/mn»«/menclose»«mprescripts/»«none/»«mn mathcolor=¨#FF0000¨»1«/mn»«/mmultiscripts»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mmultiscripts»«menclose notation=¨updiagonalstrike¨»«mn»2«/mn»«/menclose»«mprescripts/»«none/»«mn mathcolor=¨#FF0000¨»1«/mn»«/mmultiscripts»«menclose notation=¨updiagonalstrike¨»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/menclose»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»2«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mrow»«mrow»«mn»2«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The simplified expression is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfrac»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mo»§#8722;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».