Algebra 1
Review- Factoring
Name_____________________________________
Date _____________________________________
Objective:
To Review Factoring of:
 GCF
 Perfect square trinomials
 Difference of two squares
 Sum of two squares *
 Factoring a Trinomial that is not a Perfect Square
Greatest Common Factor
To factor a polynomial with a GCF follow the steps below:
1) Identify the GCF
2) Divide each term by the GCF
3) Place the GCF in front of the resulting quotient
4) Check to see if remaining expression(s) is/are factorable.
factor : 9 x 4  27 x 2
Example 1:
9 x 2 ( x 2  3)
Answer:
Perfect Square Trinomial
To factor a perfect square trinomial follow the steps below:
1) Create two empty binomials as indicated on the right
2) Take the square root of the first term of the given trinomial
3) Take the square root of the last term of the given trinomial
4) Take the result of step 2 and put in the 1st position in each binomial
5) Take the result of step 3 and put in the 2nd position in each binomial
6) The signs in the binomials should be the same as the middle term of the binomial
(
)(
)
factor : 4 x 2  12 x  9
Example 2:
Answer:
(2x - 3)2
a b
Difference of Two Squares
To factor a difference of two squares follow the steps below:
1) Create two empty binomials as indicated on the right
(
)(
)
2) Take the square root of the first term of the given binomial and put in the 1 st position of each binomial
3) Take the square root of the last term of the given binomial and put in the 2nd position of each binomial
4) Make one binomial a sum and the other binomial a difference
2
2
9 x y  16
Example 3:
Answer: (3xy - 4) (3xy + 4)
2
2
a b
Sum of Two Squares
The sum of squares is not factorable
2
2
Other Trinomials of the form: ax  bx  c
To factor a trinomial that is not one of the cases above follow the steps below:
1) List all of the factors of the “c” term
2) Find the pair of factors that add to the “b” term
3) Create two binomials
2
x  5x  6
Example 4:
Answer: x  3 x  2
2



1
Algebra 1
Lesson- Simplifying Rational Expressions
& Undefined Rational Expressions
Objective-
Name_____________________________________
Date _____________________________________
To learn how to reduce rational expressions to lowest terms.
Do Now:
On your calculator, graph the following: For each one, find the values of x for which the function is
undefined (Hint: Look for the y-value showing as “ERROR”)
Values of x for which the function is not defined:
1.
2.
2
x
5
y
x 1
y
3. y 
7x
2x  1
__________
__________
__________
Using the examples above, can you determine a way to find the values of x without graphing?
1.
y
2
x
2. y 
5
x 1
3. y 
7x
2x  1
Now try these: (Find the value of x that makes the expression undefined):
4.
2x  1
2x  4
5.
x2  2
x2  x  2
Note: Factor the denominator, set it = to zero, and solve the equation
6.
3x  4
x  3x  4
2
2
Reducing a rational expression - to reduce (or simplify) a rational expression means to write the answer in
lowest terms. To do this, factor the numerator if it can be factored, factor the denominator if it can be factored,
and then cancel any like factors.
Examples. Reduce to lowest terms:
1.
x4
3 x  12
2.
x 2  3x  4
x 1
3.
x2  9
x3
4.
2x  x 2
x2  x  2
5.
x 2  2x  3
x 2  5x  4
6.
x 2 y  xy2
xy
7.
12  4a
a  a  12
2
Now put it all together:
1) Determine the value(s) that make the following rational expressions undefined. Then simplify each.
x
x2  9
a)
b)
3
x  16 x
x3
3
Algebra 1
Lesson- Operations with Rational Expressions
Name_____________________________________
Date _____________________________________
Objective:
To know how to Add, Subtract, Multiply and Divide Rational Expressions
DO NOW:
Factor each rational expression completely:
a)
h 2  2h  3
h2  9
Suppose that we wanted to multiply those rational expressions:
b)
h 2  5h  6
h2 1
h 2  2h  3 h 2  5h  6

h2  9
h2  1
How could we use the simplified form of the fractions to do that?
Multiplying Rational Expressions:
Steps:
 Factor each numerator and denominator completely
 Cancel any like factor in any numerator with any like factor in any denominator
 Multiply the remaining expressions in each numerator
 Multiply the remaining expressions in each denominator
 Reduce if possible
Examples:
y2  y y2  4

1.
y  2 y 2  3y
2.
2 x 2  10 x x 2  5 x  6

x 2  2 x x 2  2 x  15
3.
2(a  b) 2 4( x  y ) 3( x  y )


3( x  y ) 2 9(a  b) 7(a  b)
4
Now try this:
x2 x2

8x
4x
What is the new step that you need to remember to do when dividing?
Dividing Rational Expressions:
Steps:
 Multiply the first fraction by the reciprocal of the second (KCF)
 Continue using the rules for Multiplying Rational Expressions:
Examples:
b2
1

1.
2
b 9 3b
2.
x 2  3x  2 x 2  7 x  12

x 2  3x
x 3  3x 2
5
Adding and Subtracting Rational Expressions
Steps:
 Factor all denominators
 Find the least common denominator among all fractions (if none already exists)
 Multiply each denominator and denominator by an appropriate factor to make it equivalent to the LCD
 Combine all numerators (make sure the signs are placed appropriately), simplify and put over LCD
 Reduce if possible
Combine each set of rational expressions and simplify.
Examples
8x  4 4x  6

1.
2x  6 2x  6
3.
x
2
 2
x  x  6 x  5x  6
2
2.
x
2x  2
 2
x  3 x  2x  3
4.
2x  1
4
 2
x  5 x  3 x  10
6
Algebra 1
Lesson- Solving Fractional Equations
Name_____________________________________
Date _____________________________________
Objective:
To learn how to solve equations involving rational expressions
Do Now:
1
1

x x 1
Solving Fractional Equations:
1) Find the LCD
2) Multiply each term of the equation (on both sides) by that LCD
3) Simplify and solve
Examples
1.
x x
  10
2 4
2.
1
1

2
x x 1
3.
10
4
5
 
x( x  2) x x  2
4.
3
1 1
 
x  2 x 5x
7