Online lecture for Chapter R - Professor Payne

1. Chapter R
Review: Algebraic Expressions
• Goals
– Factor a polynomial
– Factor and simplify rational expressions
– Simplify complex fractions
– Rationalize an expression involving radicals
– Simplify an algebraic expression with negative
exponents

2. Factoring Polynomials
The first few problems in your Chapter R
homework assignment have to do with
factoring. Please take advantage of the built-in
help in MyMathLab (Help Me Solve This, View
an Example, the link to the eBook) if you need
assistance with these problems. Also, feel free
to contact me if you need additional assistance.
This is a critical skill in this course and in
Calculus.

3. Rational Expressions
• A rational expression is a quotient, , of
two polynomials and .
• 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.
( )
( )
p x
q x
( )p x ( )q x

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. Another Example
• Simplify the expression:
−
−
3
4 12
x
x
Solution:
3 3
4 12 4( 3)
Note that ( ) .
Thus, 3 (3 ).
3 3
So we have
4( 3)
x x
x x
a b b a
x x
x x
x
− −
=
− −
− = − −
− = − −
− −
=
− 4 (3 )x× − −
1
.
4
= −
 
 

6. Domain and Examples
• The domain of a rational expression
consists of all real numbers except those that
make the denominator zero…
– …since division by zero is never allowed!
• Examples:
( )
( )
p x
q x

7. Video Examples: Factoring
Please view the video linked below for an
overview of factoring, leading up to a more
complicated, Precalculus-level, factoring
problem.
http://create.lensoo.com/watch/b55K

8. 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

9. Solution, continued
5 52 23 3 3
3
11
44
33
33
x x
xx
x x x
x xx
 
− × × − × ÷
  =
  × + ×+ × ÷
 
5
x
=
3 x× 2 3
1
3
x
x
− ×
3x×
15 6
1 124 3
x
xx
−
=
++ ×

10. 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

11. Example: Simplify
−
− −
−
+
2 5
3 2
x y
x y
The previous page in Blackboard contained the
following YouTube video with a solution to the
example problem above:
https://youtu.be/4O-2V1G18Qc

12. 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

13. 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

14. Example
• Rationalize the denominator of
• Note that we can also rationalize numerators
in the same way!
+
1
x y

15. 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 squar
.
.
Multiply out the d
es
enominator.
b c b c
b c b c
b c
b c b
b c
b
c
c
− −
×
+ −
−
+ −
=
−
−
=
( )( ) bc
b
b c
c
+ −
−
( )( )b c b c= + −

16. Example
Write the expression as a single quotient in
which only positive exponents and/or radicals
appear:
Assume that
Video solution: https://youtu.be/ecDpNBG3R-U
(contained on previous slide in Blackboard)
( ) ( )
−
+ − +
+
1/2 1/2
5 7 5
5
x x x
x
5.x > −

17. 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
Written solution to previous
example:

18. 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
−
+

19. ( ) ( ) ( )
( ) ( ) ( )
− − − − −−
−
 + + − +
 
+
 + + − +
 =
+
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

20. 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 >

21. 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

22. ⋅   
+ = +   ⋅   
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