11-1 Simplifying Rational Expressions

Quick Review

A rational expression is an expression that can be written in the form polynomial polynomial . fraction polynomial , over polynomial end fraction . .  A rational expression is in simplified form when the numerator and denominator have no common factors other than 1.

Example

What is the simplified form of x 2 − 9 x 2 − 2 x − 15 ? fraction x squared , minus 9 , over x squared , minus 2 x minus 15 end fraction . question mark

x 2 − 9 x 2 − 2 x − 15 = ( x + 3 ) ( x − 3 ) ( x + 3 ) ( x − 5 ) Factor the numerator and denominator .   = ( x + 3 ) 1 ( x − 3 ) ( x + 3 ) 1 ( x − 5 ) Divide out the common factor .   = x − 3 x − 5 Simplify . table with 3 rows and 3 columns , row1 column 1 , fraction x squared , minus 9 , over x squared , minus 2 x minus 15 end fraction , column 2 equals . fraction open , x plus 3 , close . open , x minus 3 , close , over open , x plus 3 , close . open , x minus 5 , close end fraction , column 3 table with 2 rows and 1 column , row1 column 1 , cap factorthenumeratorand , row2 column 1 , denominator . . , end table , row2 column 1 , , column 2 equals . fraction cross out enclosing , open , x plus 3 , close , end crossout to the first . open , x minus 3 , close , over prescript , . lower , 1 with base cross out enclosing , open , x plus 3 , close , end crossout . . open , x minus 5 , close end fraction , column 3 table with 2 rows and 1 column , row1 column 1 , cap divideoutthecommon , row2 column 1 , factor , . , end table , row3 column 1 , , column 2 equals . fraction x minus 3 , over x minus 5 end fraction , column 3 cap simplify , . , end table

Exercises

Simplify each expression. State any excluded values.

4. 2 x 2 + 6 x 10 x 3 fraction 2 , x squared , plus 6 x , over 10 , x cubed end fraction
5. m − 3 3 m − 9 fraction m minus 3 , over 3 m minus 9 end fraction
6. x 2 + 6 x + 9 5 x + 15 fraction x squared , plus 6 x plus 9 , over 5 x plus 15 end fraction
7. 2 a 2 − 4 a + 2 3 a 2 − 3 fraction 2 , eh squared , minus 4 eh plus 2 , over 3 , eh squared , minus 3 end fraction
8. 2 s 2 − 5 s − 12 2 s 2 − 9 s + 4 fraction 2 , s squared , minus 5 s minus 12 , over 2 , s squared , minus 9 s plus 4 end fraction
9. 4 − c 2 c − 8 fraction 4 minus c , over 2 c minus 8 end fraction
10. Geometry What fraction of the rectangle is shaded? Write your answer as a rational expression in simplified form.

    

11-2 and 11-3 Multiplying and Dividing Rational Expressions and Dividing Polynomials

Quick Review

You can multiply and divide rational expressions using the same properties you use to multiply and divide numerical fractions.

a b ⋅ c d = a c b d , eh over b , dot , c over d , equals . fraction eh c , over b d end fraction . comma  where b ≠ 0 b not equal to 0  and d ≠ 0 . d not equal to 0 .

a b ÷ c d = a b ⋅ d c = a d b c , eh over b , divides , c over d , equals , eh over b , dot , d over c , equals . fraction eh d , over b c end fraction . comma  where b ≠ 0 , c ≠ 0 , b not equal to 0 comma c not equal to 0 comma  and d ≠ 0 . d not equal to 0 .

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. To divide a polynomial by another polynomial, use long division. When dividing polynomials, write the answer as quotient + remainder divisor . plus . fraction remainder , over divisor end fraction . .

Example

What is the quotient y y + 3 ÷ y − 3 y − 2 ? fraction y , over y plus 3 end fraction . divides . fraction y minus 3 , over y minus 2 end fraction . question mark

y y + 3 ÷ y − 3 y − 2 = y y + 3 ⋅ y − 2 y − 3   = y ( y − 2 ) ( y + 3 ) ( y − 3 ) table with 2 rows and 2 columns , row1 column 1 , fraction y , over y plus 3 end fraction . divides . fraction y minus 3 , over y minus 2 end fraction , column 2 equals . fraction y , over y plus 3 end fraction . dot . fraction y minus 2 , over y minus 3 end fraction , row2 column 1 , , column 2 equals . fraction y . open , y minus 2 , close , over open , y plus 3 , close . open , y minus 3 , close end fraction , end table

Exercises

Multiply or divide.

11. 4 x + 12 x 2 − 2 x ⋅ x 6 x + 18 fraction 4 x plus 12 , over x squared , minus 2 x end fraction . dot . fraction x , over 6 x plus 18 end fraction
12. a 2 + 5 a + 4 a 3 ÷ a 2 + 3 a + 2 a 2 − 2 a fraction eh squared , plus 5 eh plus 4 , over eh cubed end fraction . divides . fraction eh squared , plus 3 eh plus 2 , over eh squared , minus 2 eh end fraction
13. x 2 + 13 x + 40 x − 7 ÷ x + 8 x 2 − 49 fraction x squared , plus 13 x plus 40 , over x minus 7 end fraction . divides . fraction x plus 8 , over x squared , minus 49 end fraction
14. ( 12 x 2 + 9 x − 7 ) ÷ 3 x open 12 , x squared , plus 9 x minus 7 close divides 3 x
15. ( 3 d 2 + 2 d − 29 ) ÷ ( d + 3 ) open 3 , d squared , plus 2 d minus 29 close divides open d plus 3 close

16. Geometry The width and area of a rectangle are shown in the figure below. What is the length of the rectangle?

    

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