Rational Expressions. Definition: P, is an algebraic expression that can be written as the quotient of two polynomials, P. A rational expression, Q

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1 Rational Eressions What is a Rational Eression? A rational eression is siml a fraction so conseuentl the are sometimes called algebraic fractions To be more secific, a rational eression is an algebraic eression that can be written as the uotient of two olnomials This definition is similar to a rational number which is a number that can be written as the uotient of two integers Definition: A rational eression, Q P, is an algebraic eression that can be written as the uotient of two olnomials, P and Q, where Q Eamle: The following are all rational eressions Notice that each numerator and each denominator is a olnomial The olnomial ma be a monomial, a binomial, a trinomial, or a olnomial Division b Zero: Because a rational eression is a fraction we need to make sure that the denominator is never eual to ero Remember that division b ero is mathematicall undefined, that is, we siml cannot do it Therefore we alwas need to restrict the values of the denominator to non-ero numbers This is most easil accomlished b setting the denominator eual to ero and solving the resulting euation The rational eression is therefore undefined at these values Eamle: Identif all the values for which the rational eression is undefined Solution: Set the olnomial in the denominator eual to ero and solve Therefore, the rational eression is undefined at This can be demonstrated b substituting this value into the original eression Notice that the numerator is not involved in this roblem at all

2 Eamle: Identif all the values for which the rational eression is undefined Solution: Set the olnomial in the denominator eual to ero and solve Therefore, the rational eression is undefined at, and - This can be demonstrated b substituting these values into the original eression Eamle: Identif all the values for which the rational eression is undefined Solution: Set the olnomial in the denominator eual to ero and solve and Therefore, the rational eression is undefined at, and Eamle: Identif all the values for which the rational eression is undefined Solution: Set the olnomial in the denominator eual to ero and solve and Therefore, the rational eression is undefined at -, and

3 Euivalent Eressions: Two rational eressions are said to be euivalent if the have the same value Recall from our work with fractions that euivalent fractions are two fractions that have the same value although the are written with different numbers Eamle: The following fractions are all euivalent Notice that the all reresent the same amount,,,, Eamle: The following rational eressions are all euivalent If we substitute an value in for that the resulting eressions will be euivalent fractions Let Finding Euivalent Eressions To find an euivalent rational eression, multil the numerator and denominator of Q P b the same olnomial R Where Q and R P Q PR QR Eamle: Find a rational eression that is euivalent to Solution: To find an euivalent eression we must multil the numerator and denominator b some value R Let R Let R

4 Eamle: Determine if the two rational eressions a and a are euivalent a a Solution: If we let R a we have, a a a a a a Therefore, the two rational eressions are euivalent Eamle: Find a rational eression that is euivalent to but has a denominator of Solution: The denominator is and Therefore if we let R we have, Therefore, the two rational eressions are euivalent Simlifing Rational Eressions: A rational eression is said to be in simlest form when its numerator and denominator have no common factor other than ± To simlif a rational eression Factor the numerator and denominator Divide Cancel out an common factors Eamle: Write in simlest form Solution: Since this involves two monomials there is nothing to factor, therefore just cancel out the common factors of

5 a bc Eamle: Write in simlest form a b Solution: Since this involves two monomials there is nothing to factor, therefore just cancel out the common factors of a bc a b a ba c a bb a c b The last two roblems could have also been solved using the eonent rules I leave it to ou to confirm this Eamle: Write in simlest form Solution: Now that we have binomials in the numerator and denominator we can do some factoring In this case we need to factor out the GCF of and then cancel divide it out ab ac Eamle: Write in simlest form a a Solution: Factor out the GCF a, then divide it out ab ac a a a b c a b c

6 Eamle: Write in simlest form Solution: Factor and divide Eamle: Write in simlest form Solution: Factor and divide Eamle: Write in simlest form Solution: Notice that the factors in the numerator and denominator are the same but with oosite signs In this situation factor out a - from both factor, and then divide

7 Eamle: Write in simlest form Solution: Factor and divide Eamle: Simlif Solution: Factor and divide Eamle: Simlif Solution: Factor and divide

8 Eamle: Simlif Solution: Factor and divide Eamle: Simlif Solution: Factor and divide

9 -Alications: Eamle: When the mathematics deartment at Grand Canon Universit had more facult and staff members, each had an office with dimensions of ft b ft Now that the deartment has gotten smaller, its offices are being enlarged Each facult office and each staff office will be made ft wider, whereas each facult office will also be made ft longer Write an eression that indicates how much larger the area of a new facult office be than a new staff office How much larger is a facult office if the original width is ft? Solution: Write an eression for the area with the formula A L W Enlarged Facult Office Area Enlarged Staff Office Area Create a ratio of the area of an enlarged facult office to enlarged staff office and simlif If the original length was ft, then the eression above becomes A facult office will be twice as large as a staff office Eamle: The force of gravit between two lanets is given b the eression kmm, d where M and m are the masses of the lanets, d is the distance between the lanets, and k is a fied constant Under what circumstances is the force of gravit undefined? Solution: The force of gravit will be undefined when the distance is ero This is determined b setting the denominator eual to ero and solving d d

10 Eamle: An eression imortant in the stud of nuclear energ is mu mv mu mv, where m reresents mass and u and v reresent velocities Write this eression in simlified form Solution: This is a matter of simlifing the rational eression b factoring and cancelling mu mv mu mv m u v u v m u v u v