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Section A.3
Rational Expressions
• Goals
– Factor and simplify rational expressions
– Simplify complex fractions
– Rationa...
Rational Expressions
• A rational expression is a quotient p/q of two
polynomials p and q .
• When simplifying a rational ...
Domain and Examples
• The domain of a rational expression p/q
consists of all real numbers except those that
make the deno...
Example
• Simplify the expression:
−
+
2
25 100
5 10
t
t
( ) ( )22
Solution:
25 4 25 2 ( 2)25 100
5 10 5( 2) 5( 2)
5 25
t ...
Another Example
• Simplify the expression:
−
−
3
4 12
x
x
Solution:
3 3
4 12 4( 3)
Note that 1( ) .
Thus, 3 1(3 ).
3 3
So ...
Simplifying a Complex Fraction
5
2
Example: Simplify .
1
4
3
x
x
−
+
To simplify a complex fraction, multiply the entire n...
Solution, continued
5 52 3 3 2 3
11
3 4 34 3
33
x x x
x x
x xx
xx
 
− × × − × ÷
  =
  × + ×+ × ÷
 
5
x
=
3 x× 2 ...
Expressions with Negative Exponents
• Recall that
• If the expression contains negative exponents,
one way to simplify is ...
Example: Simplify
−
− −
−
+
2 5
3 2
x y
x y
−
− −
−
−
=
+
+
2
2 5 5
3 2
3 2
1
1 1
x
x y y
x y
x y
The least common denomin...
Rationalizing the Denominator
• Given a fraction whose denominator is of the
form or we sometimes want
to rewrite the frac...
Rationalizing (cont’d)
• To rationalize the denominator, multiply both
numerator and denominator of the fraction by
the co...
Example
• Rationalize the denominator of
• Note that we can also rationalize numerators
in the same way!
+
1
x y
Another Example
• Rationalize the denominator of
−
+
2 2
b c
b c
( )( )
( )( )
( )
2 2
2 2
Multiply numerator and denomina...
Example
Write the expression as a single quotient in
which only positive exponents and/or radicals
appear:
Assume that
( )...
First, notice that the numerator has a common
factor of x + 5.
Always take the lower exponent when taking
out a common fac...
To find the exponent that is left when you take
out a common factor to a certain power,
subtract that power from each expo...
( ) ( ) ( )
( ) ( ) ( )
− − − − −−
−
 + + − +
 
+
 + + − +
 =
+
1/2 ( 1/2) ( 1/21/2 1/2
1/2 1 0
)
5 5 7 5
5
5 5 7 ...
Example
Write the expression as a single quotient in
which only positive exponents and/or radicals
appear:
Assume that
−
+...
Again, factor out the common factor with the
lower exponent:
Now combine the expressions to form a single quotient.
− − − ...
×   
+ = + ÷  ÷×   
1 4 1 2 3 4
2
3 1 3 3
x x
x x
+ + 
= = ÷
 
1 6 4 6 4
3 3
x x
x x
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A 3rationalexpressions-120114133135-phpapp02

  1. 1. Section A.3 Rational Expressions • Goals – Factor and simplify rational expressions – Simplify complex fractions – Rationalize an expression involving radicals – Simplify an algebraic expression with negative exponents
  2. 2. Rational Expressions • A rational expression is a quotient p/q of two polynomials p and q . • When simplifying a rational expression, begin by factoring out the greatest common factor in the numerator and denominator. • Next, factor the numerator and denominator completely. Then cancel any common factors.
  3. 3. Domain and Examples • The domain of a rational expression p/q consists of all real numbers except those that make the denominator zero… – …since division by zero is never allowed! • Examples:
  4. 4. Example • Simplify the expression: − + 2 25 100 5 10 t t ( ) ( )22 Solution: 25 4 25 2 ( 2)25 100 5 10 5( 2) 5( 2) 5 25 t t tt t t t − − +− = = + + + = ( )2 ( 2)t t− + 5 ( 2)t + 5( 2)t= − (Solution will appear when you click)
  5. 5. Another Example • Simplify the expression: − − 3 4 12 x x Solution: 3 3 4 12 4( 3) Note that 1( ) . Thus, 3 1(3 ). 3 3 So we have 4( 3) x x x x a b b a x x x x x − − = − − − = − − − = − − − − = − 4 1(3 )x×− − 1 . 4 = −
  6. 6. Simplifying a Complex Fraction 5 2 Example: Simplify . 1 4 3 x x − + To simplify a complex fraction, multiply the entire numerator and denominator by the least common denominator of the inner fractions. The inner fractions are and . Their LCD is Thus, we should multiply the entire numerator and denominator by 5 x 1 3x 3 .x 3 .x
  7. 7. Solution, continued 5 52 3 3 2 3 11 3 4 34 3 33 x x x x x x xx xx   − × × − × ÷   =   × + ×+ × ÷   5 x = 3 x× 2 3 1 3 x x − × 3x× 15 6 1 124 3 x xx − = −+ ×
  8. 8. Expressions with Negative Exponents • Recall that • If the expression contains negative exponents, one way to simplify is to rewrite the expression as a complex fraction. • Recall the following properties of exponents: − = 1n n x x + − ⋅ = =and a a b a b a b b x x x x x x
  9. 9. Example: Simplify − − − − + 2 5 3 2 x y x y − − − − − = + + 2 2 5 5 3 2 3 2 1 1 1 x x y y x y x y The least common denominator of the inner fractions is 3 5 .x y + × − × − = = × + × + 3 5 2 3 5 3 5 2 3 5 5 5 3 5 3 5 3 5 3 5 3 2 3 2 1 1 1 x y x x y x y x y y y x y x y x y x y x y x y − = 3 5 5 5 x y x y 5 y 3 x 5 3 y x − − = + + 5 5 3 5 3 3 3 5 2 x y x y x y x y First, rewrite the expression as a complex fraction. Multiply numerator and denominator by the LCD.
  10. 10. Rationalizing the Denominator • Given a fraction whose denominator is of the form or we sometimes want to rewrite the fraction with no square roots in the denominator. • This is called rationalizing the denominator of the given fractional expression. • It often allows the fraction to be simplified. a b+ +a b
  11. 11. Rationalizing (cont’d) • To rationalize the denominator, multiply both numerator and denominator of the fraction by the conjugate of the denominator. • The conjugate of is the conjugate of is a b+ ;a b− + .a b−a b
  12. 12. Example • Rationalize the denominator of • Note that we can also rationalize numerators in the same way! + 1 x y
  13. 13. Another Example • Rationalize the denominator of − + 2 2 b c b c ( )( ) ( )( ) ( ) 2 2 2 2 Multiply numerator and denominator by the conjugate of the denominator. Factor the difference of squares. Multiply out the denominator. b c b c b c b c b c b c b c b c b c − − × + − − − = + − − = ( )( )b c b c b c + − − ( )( )b c b c= + −
  14. 14. Example Write the expression as a single quotient in which only positive exponents and/or radicals appear: Assume that ( ) ( ) − + − + + 1/2 1/2 5 7 5 5 x x x x 5.x > −
  15. 15. First, notice that the numerator has a common factor of x + 5. Always take the lower exponent when taking out a common factor. In this case, the exponents are 1/2 and -1/2, so the lower exponent is -1/2. ( ) ( ) − + − + + 1/2 1/2 5 7 5 5 x x x x
  16. 16. To find the exponent that is left when you take out a common factor to a certain power, subtract that power from each exponent of the common factor. Remember, we are factoring out continued on next slide. . . ( ) ( ) ( ) − − − − −−  + + − +   + 1 1/2 1/2/2 ( 1/2) ( 1/2) 5 5 7 5 5 x x x x x ( ) 1/2 5 .x − +
  17. 17. ( ) ( ) ( ) ( ) ( ) ( ) − − − − −− −  + + − +   +  + + − +  = + 1/2 ( 1/2) ( 1/21/2 1/2 1/2 1 0 ) 5 5 7 5 5 5 5 7 5 5 x x x x x x x x x x ( ) ( ) + − = + + 1/2 5 7 5 5 x x x x − = + 3/2 5 6 ( 5) x x
  18. 18. Example Write the expression as a single quotient in which only positive exponents and/or radicals appear: Assume that − +1/2 1/24 2 3 x x 0.x >
  19. 19. Again, factor out the common factor with the lower exponent: Now combine the expressions to form a single quotient. − − − − −− −  + = + ÷   1/2 (1/2 1/2 1/2 11 //2) ( 1/2 2)4 4 2 2 3 3 x x x x x 1/2 x− − −    = + = + ÷  ÷     1/2 0 1 1/24 4 2 2 3 3 x x x x x   = + ÷   1 4 2 3 x x
  20. 20. ×    + = + ÷  ÷×    1 4 1 2 3 4 2 3 1 3 3 x x x x + +  = = ÷   1 6 4 6 4 3 3 x x x x

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