College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran
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1 College Algebra - MAT 6 Page: Copyright 009 Killoran Rational Expressions Fractional Polynomial Expressions would be another appropriate title for this section. When working with these Rational Expressions it is best to understand the rules governing rational numbers. How do you simplify fractions, multiply fractions, divide fractions, and add fractions? Given your mastery (or lack) of these answers will directly influence your understanding of this section. Simplification (lowest terms) To reduce a fraction it is only necessary to remove the common factor from the numerator and the denominator. For instance: 4 : To simplify a rational expression we will need to factor the parts to find common factors. Example Simplify the Rational Expression: C x 7x C To start with we will factor the top and bottom individually. So factor C x first C x C 6x x x.x C /.x C /.x C /.x / Then factor 7x C 7x C 6x x C x.x /.x /.x /.x / Now putting them back into place C x.x C /.x / 7x C.x /.x / x C x Now there is a cause for concern with these fractions, remember that the denominator cannot be zero.
2 College Algebra - MAT 6 Page: Copyright 009 Killoran So we must make sure that.x /.x /, the original denominator, is not zero. This is known as finding the domain of the expression. The first factor, x 6 0, which implies that x 6 and the second factor, x 6 0; which implies that x 6 =: Thus domain of this fraction, taken before reducing, is All Real Numbers.R/ except, x 6 f; =g: Written as: x 6 f; =g Example Simplify the rational expression: 8 7 C x 8 7 C x 8 9 C x 8.x /.x C /.x C /.x / 8.x C / x C The omain would be where.x C /.x / 6 0; thus x 6 f ; g Multiplication of Rational Expressions: To multiply fractions: a b c d ac bd by recognizing the fact that 4 and 6 have common factors as well as and. We can use the same idea with Rational Expressions: Example Simplify the rational expression: C 4 C 4x C 4x C 9x C 8 Working individually with each individual fraction we factor each piece: C 4 C 4x C.x C 6/./.x C /.x C / 4x.x C /.x / C 9x C 8.x C 6/.x C /
3 College Algebra - MAT 6 Page: Copyright 009 Killoran Putting them back into order and we get:.x C 6/./.x C /.x /.x C /.x C /.x C 6/.x C / Removing any factors that are common from the anywhere on the top to the any on the bottom and we get: x+6fi x? fi x+fi x? fi 6 x + fi x+fi x+6fi x + fi./.x /.x C / and we have to remember that x 6 f ; ; 6g Example 4 Find the product of these rational expressions: x C 6 7x C 0 4 C 7x 6 x C 6 7x C 0 4./.x / C 7x 6./.x /./.x C /.x C /./ x 6.x /.x C /.x /.x C / ; ; ; ivision of Rational Expressions: ivision between fractions involves multiplication of the reciprocal of the second fraction (or the fraction in the denominator) a b c d a b d c
4 College Algebra - MAT 6 Page: 4 Copyright 009 Killoran Example Find the quotient of these rational expressions: C 6x 7 x C 7 8x C 7 C 6x 7 x C 7 8x C 7 C 6x 7 x 8x C 7 x C 7.x /.x 7/.x C 7/.x / x C 7./.x 7/.x C 7/ x 6 f 7; g 4 Adding and Subtracting: We need to remember that when adding fractions, the fractions must first have a common denominator. **Please check first if you are indeed adding/subtracting before you get a common denominator. I always have someone that gets a common denominator for multiplication!! a b C c d a b d C c d d b ad C cb b bd So the trick in English is: "Multiply top and bottom of the fraction by the missing factors from the other denominator" Example 6 Perform the indicated operation and simplify: 4 x 4 Notice that they already have a common denominator. Write./ once and combine the "Tops"... 4 x 4./.x C / x C
5 College Algebra - MAT 6 Page: Copyright 009 Killoran Example 7 Perform the indicated operation and simplify: x x C The missing factor on the Left is.x C / so we will multiply the top and bottom of the first Rational Expression by.x C / : The missing factor on the Right is.x / so we will multiply top and bottom of the second Rational Expression by.x /. After this they will have a common denominator of.x /.x C / and we will only need to write it once in the denominator and combine what is on the top. x x C x? fi 6 x + fi x + fi? x + fi 6 x? fi x? fi.x C /.x /.x /.x C / x C 7.x /.x C / Remark Leave the distributive property until you write the entire top on one fraction line. That way the distribution is done correctly. Mistakes are easily made when doing the distribution in your head! Example 8 Perform the indicated operation and simplify: x C 4 6x C Sometimes it will be necessary to factor the denominators to find the "missing" factors. x C 4 6x C.x C / 6.x C /.x C / 6.x C /.nothing/ 9 6.x C / 6.x C / 7 6.x C /
6 College Algebra - MAT 6 Page: 6 Copyright 009 Killoran Example 9 Simplify the rational expression: x x x C 6 x.x C /.x / x x.x /./ x C 6.x /./.x C /.x /././.x C /.x /./.x C /.x /./.x C /.x C /.x /./ x x C x.x C /.x /./ 8x.x C /.x /./ Complex Fractions (fractions within a fraction) Example 0 Simplify the Complex Fraction: 4 6 C Method I Finding the common denominator for the top and then the bottom we get to the "answer" C 6 C C Method II A nice trick if you want to reduce the amount of work required to reduces a fraction. Here we will multiply each term by the common denominator of all the fractions C C 6 0 C The step with the times everything, can be done in your head. For instance 6 can be done by finding first and then multiplying by : 6
7 College Algebra - MAT 6 Page: 7 Copyright 009 Killoran Example Simplify the following complex fraction: x x x C x The common denominator here will be [x.x /] : So we will multiply all terms top and bottom by this: x.x / x x x x x C x x C x.x / x x x.x /.x / x.x / C x x.x / x.x / x.x / x x? fi x x x? fi x x? fi +? x x? fi x x? fi x x? fi x.x /.x / C x x x C 4x
8 College Algebra - MAT 6 Page: 8 Copyright 009 Killoran Optional: An advanced example that is used in Calculus. p px x C Example Rationalize the Numerator p p! x x C p px p px x C x C p p./ x x C p p x x C p p./ x x C p p x x C p p./ x x C p x C p x C p x C p x C x.x C / x p x C C.x C / p x x p x C C.x C / p x x p x C C.x C / p x
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