SAT Prep
A.) Vocabulary
Monomials – Any number or variable or product of number(s)
and variable(s)
Ex. Evaluate 3a 2 b when a = -4 and b = 0.5.
3  4   0.5  
2
3 16 0.5 
24
Binomials – 2 monos separated by +/-
2x  3
Trinomials – 3 monos separated by +/-
2 x2  3x 1
B.) Simplifying Polynomials
!!** ADD OR SUBTRACT LIKE TERMS ONLY
**!!
Like terms – same variable(s) and same exponent(s).
Ex. Simplify 3 x 2  4 x  2 x 2  5 x.
2
2
3
x

2
x

  4x  5x  
5x2  9 x
Multiplying Monomials - Mult. Coeffecients and add
exponents of like bases
Ex. Simplify the following: (3 xy z )(2 x y)
2 3
 3  2   x  x
2
 y  y  z
2
3

2
 6 x y z
3
3 3
Dividing MonomialsDiv. coeff. And subtract
exponents of like bases
(6 x 4 y 2 z 3 )
Ex. Simplify the following: (2 x 2 z )
6  x 4  y 2  z 3 
  2   
2  x  1  z 
3  x 42  y 2  z 31 
 
 

1  1  1  1 
 3 x 2 y 2 z 2
C.) Factoring and Expanding
FOIL –
First
Outer
Inner
Ex . Expand the following:
Last
 x  33x  7 
FIRST:
OUTER:
INNER:
x  3x
x  7 
3  3x 
LAST:
3  7 
3x 2
7 x
9 x
21
3 x 2  2 x  21
Three important binomial products
( x  y )( x  y )  x 2  y 2
( x  y )  x  2 xy  y
2
2
2
( x  y ) 2  x 2  2 xy  y 2
Ex. If (a – b) = 17.5 and (a + b) = 10, what does a2 – b2 =?
a2  b2   a  b  a  b 
 a2  b2  17.510 
175
Ex. If x2 + y2 = 36 and (x + y) 2 = 64 what is xy?
 x  y
2
 x 2  2 xy  y 2
64  x 2  2 xy  y 2
64  2 xy  36
28  2xy
xy  14
D.) MORE FACTORING
GCF, Common Monomials, and Product/sum table
2
Ex. Find all real solutions of x  x  6  0
 x  2 x  3  0
FACTORS (-6) SUM (-1)
1, 6
5
1, 6
2, 3
5
2, 3
1
1
 x  2  0
or
 x  3  0
x  2 or x  3
2
3
x
Ex. Find an equivalent expression for 2  12 .
x  4x  4
3 x2  4
 2
x  4x  4
3 x  2

3  x  2  x  2 
 x  2

 x  2  x  2 


1
1
Ex. Find the sum of reciprocals of 2 and 2 .
x
y
x2 + y2
A.) Single Equations
Ex. Solve the following for x:
1
x  3( x  2)  2( x  1)  1
2
1
x  3x  6  2 x  2 1
2
1

2  x  3x  6  2 x  3
2

x  6 x 12  4 x  6
7 x 12  4 x  6
3x  18
x6
Ex. If a = b(c + d), solve for d in terms of a, b, and c.
a  bc  bd
a  bc  bd
a  bc
d
b
a  bc
a
d
or d   c
b
b
Ex. If 3 x  1  5 , then x = ?
3 x 6
x4
x 2
 x
2
  2
2
Ex. If 2x – 5 = 98, then 2x + 5 = ?
2 x  5  2 x  5 10
108
2 x  5  98 10
Ex. For what value of x is
3 6

5 x
3x  30
4 3 10
?
 
x 5 x
10
Ex. If
y  5 x 2  3 , solve for x.
y  3  5x2
y 3
 x2
5
y 3
x
5
Ex. If x 2  4  125 , then x = ?
x 2  121
x 2  121
x  11
2
2
x
 3x  0 .
Ex. Find the largest value of x that satisfies
x  2 x  3  0
x  0 or
 2 x  3  0
3
x
2
w 3
w 1
4

8
Ex. If
, what is w?
2 
2 w 3
2
2 w 3
 2

3 w 1
  2
3 w 1
2w  6  3w  3

2  w  3  3  w  1
w 9
B.) Systems of Equations/Inequalities
Use appropriate method
Substitution, elimination, graphing, matrices
Ex. Solve for x and y if x + y = 10 and x – y = 2.
x  y  10
 x y2
2 x  12
x6
6  y  10
y4
 6, 4 
Ex. If 3a + 5b = 10 and 5a + 3b = 30, what is the average of a
and b?
3a  5b  10
 5a  3b  30
8a  8b  40
8
ab 5
ab 5

2
2
5
2
READ, READ, READ, READ, and READ AGAIN!!!
A.) Strategies
1.) Substitution i.e., “Plugging it in”
Why???
- Numbers make more sense than letters.
- Choose numbers easy to work with, but not 0 and 1.
- 2,3,5, etc. are good choices for algebra problems.
- Multiples of 100 for percent problems.
- Multiples of 60 for time problems.
When???
-You have NO idea how to do the problem
-There is a variable in the question and the
answers are all numbers
-The problem is about “some number” and you
have no clue as to what that number is.
B.) Examples
Ex. The price of an item in a store is d dollars. If the sales tax
on the item is s%, what is the total cost of x such items,
including tax?
a.) xds
b.) xds  1
xd  s  1
c.) 100
d.) 100 x(d  ds )
e.) xd ( s  100)
100
Let’s choose some numbers for d, s, and x.
d  10
s  5%
x  10
The total price for 1 item = 10  10(.05)  $10.50
The total price for 10 items = 10  $10.50  $105.00
Which choice gives us $105.00? – Start with A
xds
 10 10 5  500
Obviously, B.) is out
NO
C.) xd  s  1  10 10  5  1  6
100
100
NO
D.) 100 x(d  ds)  100 10 10  105   25000 NO
xd ( s  100)
E.)
100
10 10  (5  100)

 105 YES!!!
100
Ex. Vehicle A travels at x miles per hour for x hours. Vehicle B
travels a miles faster than Vehicle A, and travels b hours
longer than Vehicle A. Vehicle B travels how much farther than
Vehicle A, in miles?
a.) x 2  ab
2
2
a

b
b.)
c.) ax  bx  ab
d.) x  abx  ab
2
e.) 2x2   a  b  x  ab
Let’s choose some numbers for x, a, and b.
Vehicle A = 20  20  400
Vehicle B =
x  20
a  10
b5
30  25  750
Vehicle B – Vehicle A = 750  400  350
By substitution – A.)
x 2  ab
 202  10(5)  350