L6 Rational Expressions - Part 1
Completion requirements
Unit 1A
Precalculus
Lesson 6: Rational Expressions
Mathematical modelling can be used in nearly every human endeavour. From science and engineering to business and music.
The formula for the length of a flute is modelled after the standing wave that forms when air inside the flute is made to vibrate.
In the formula, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»L«/mi»«/mstyle»«/math» is the length of the instrument, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»v«/mi»«/mstyle»«/math» is the speed of sound, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«/mstyle»«/math» is the frequency of the note. Without proper mathematical models of this nature, design and construction of musical instruments would be very difficult.
The formula for the length of a flute is modelled after the standing wave that forms when air inside the flute is made to vibrate.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»L«/mi»«mo»=«/mo»«mfrac»«mrow»«mn»2«/mn»«mi»v«/mi»«/mrow»«mi»f«/mi»«/mfrac»«/mrow»«/mstyle»«/math»
In the formula, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»L«/mi»«/mstyle»«/math» is the length of the instrument, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»v«/mi»«/mstyle»«/math» is the speed of sound, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«/mstyle»«/math» is the frequency of the note. Without proper mathematical models of this nature, design and construction of musical instruments would be very difficult.
Like many mathematical models, the formula governing the length of a
flute is a rational function. The right side of the formula is a
rational expression. Rational expressions are essentially fractions that
contain variables. It is important to be
confident with simplifying fractions in order to simplify rational
expressions and solve rational equations.
A rational expression is a fraction where both the numerator and denominator are polynomials. Any rational expression can be written in the form
where both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»Q«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math» are polynomials, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»Q«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
Variables used in rational expressions can have restrictions. These restrictions occur when the variables are found in the denominator of the rational expression. These restrictions, also called non-permissible values (NPVs), are any value(s) of the variable(s) that will cause the denominator to equal zero.
Simplifying a rational expression means to reduce it to lowest terms. The steps to follow when simplifying rational expressions are as follows.
A rational expression is a fraction where both the numerator and denominator are polynomials. Any rational expression can be written in the form
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»P«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«mrow»«mi»Q«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»,
where both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»Q«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math» are polynomials, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»Q«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
Variables used in rational expressions can have restrictions. These restrictions occur when the variables are found in the denominator of the rational expression. These restrictions, also called non-permissible values (NPVs), are any value(s) of the variable(s) that will cause the denominator to equal zero.
Simplifying a rational expression means to reduce it to lowest terms. The steps to follow when simplifying rational expressions are as follows.
Step 1:
Factor both the numerator and the denominator.
Step 2:
Identify any non-permissible values (NPVs) by determining the value(s) of the variable(s) that cause the denominator to equal zero.
Step 3:
Simplify by eliminating factors found in both the numerator and denominator. State the non-permissible values as part of the final simplified expression.
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