Math Analysis
Review Packet Notes I
Name: ____________________________
SOLVING EQUATIONS AND INEQUALITIES
You must have a firm grasp on how to solve all types of equations. I expect that you
already know how to solve one, two, and multi-step equations and will not provide
notes on them here. Also, be sure you show each step of your work. NO WORK, NO
CREDIT!
Inequalities
Don’t forget about working with inequalities. You need to remember to flip the
inequality symbol when dividing by a negative.
EX]
2x  -18
x  -9 DON’T FLIP
EX]
-2x  -18
x  9
FLIP
Absolute Value Equations
EX]
2x + 3 = 5
2x + 3 = 5
2x = 2
x = 1 AND
AND
AND
2x + 3 = -5 Remember to set up 2 equations!
2x = -8 Subtract 3 from both sides
x = - 4 Divide each side by 2
Radical Equations
EX]
3
8x  3  5   2
Original Problem
3
8x  3  3
Isolate radical by adding 5 to both sides
(3 8x  3 )3  (3)3
8x + 3 = 27
8x = 24
x=3
Cube each side
Simplify
Subtract 3 from both sides
Divide both sides by 8
NOTE: Be sure that you check your solution(s)! This is one of the best ways
to catch yourself making those silly little careless mistakes that drive you
and your math teachers crazy! In the case of radical equations, your
solutions may be extraneous.
Math Analysis; updated 6/19/09
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SYSTEMS OF EQUATIONS
You should know how to solve a system of equations in two-variables by graphing,
substitution, linear combination (also called elimination), and matrices. Systems
involving more than two variables are done using the calculator.
NOTE:
When



solving systems, you can get one of the following answers:
1 solution (written as a coordinate point)
infinitely many solutions
no solution.
SOLVING QUADRATIC EQUATIONS
This single concept is the bulk of Algebra II. The terms roots, zeros, x-intercepts, and
solutions are all synonymous. You should know how to solve quadratics by factoring,
completing the square, and of course, the quadratic formula (please, please, please,
memorize this!)
x
The quadratic formula:
EX]
EX]
b  b2  4ac
2a
y = x2 – x – 6
y = (x + 2)(x – 3)
Factor
x+2=0
x = -2
Set each factor equal to 0
Solve
x –3=0
x=3
y = x2 – x – 6
x
x
Let a = 1, b = -1, and c = -6
1  (1)2  4(1)(6)
Substitute into quadratic formula
2(1)
1
x
1  24
1  25
=
2
2
1 5 6
 3
2
2
AND
=
x
1 5
2
1  5 4

 2
2
2
Simplify
Simplify
DON’T FORGET TO CHECK YOUR ANSWERS!
Math Analysis; updated 6/19/09
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POLYNOMIAL FUNCTIONS
The Rational Zero Theorem
When solving polynomial equations, it is important to know the rational zero
theorem:
If f(x) = anxn + … + a1x + a0 has integer coefficients , then every rational zero has the
form
factors of constant term a0
p

q factors of leading coefficient an
EX]
Find the rational zeros of f(x) = 3x3 – 4x2 – 17x + 6
Possible rational zeros:
p
factors of constant term, 6
±1,  2,  3,  6


q factors of leading coefficient, 3
1,  3
Simplifying yields the following possibilities:
 1,  2,  3,  6, 
1
2
,
3
3
We then use either the calculator (look in the TABLE) or synthetic division to test the
possible zeros. You may have to test several before you find one that works. Here, x
= -2 works:
-2
3
3
-4
-6
-10
-17
20
3
6
-6
Remember how to get the
factor of f(x) from here?
0
Since x = -2 is a zero, then x + 2 is a factor of f.
f(x) = (x + 2)(3x2 – 10x + 3)
Factor original polynomial
f(x) = (x + 2)(3x – 1)(x – 3)
Factoring the trinomial
Setting each factor equal to 0 and solving gives us the zeros.
x+2=0
x = -2
3x – 1 = 0
x–3=0
3x = 1
x=3
The zeros are x = -2, x =
Math Analysis; updated 6/19/09
1
, and x = 3
3
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Factor by Grouping
Another skill to use when finding rational zeros of a polynomial function is factor by
grouping.
f(x) = 2x3 + 2x2 – 8x – 8
Original equation
f(x) = 2x2(x + 1) – 8(x + 1)
Factor GCF’s
f(x) = (2x2 – 8)(x + 1)
Rewrite
f(x) =2(x2 – 4)(x + 1)
Factor
The zeros are –2, 2, and –1.
NOTE:
Be careful here! A common mistake is to forget the 
x2 – 4 = 0
x2 = 4
x=2
Writing Polynomial Equations from Zeros
Given the zeros of a polynomial function, you should be able to write the equation of
the original polynomial equation. The main thing here is to watch your signs!
EX] The zeros of a polynomial function are –1, 5, and 6. Write the function of least
degree.
f(x) = (x + 1)(x – 5)(x – 6)
f(x) = (x2 – 4x – 5) (x – 6)
Use FOIL with first two terms
f(x) = x3 – 4x2 – 5x – 6x2 + 24x + 30
Multiply
f(x) = x3 – 10x2 + 19x + 30
Combine like terms
NOTE:
Use the TI-83 to check
Math Analysis; updated 6/19/09
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Put original in Y1 and your answer in Y2
Look at your table of values –
They should be the same!
RATIONAL EXPRESSIONS
I expect that you already know how to simplify rational expressions, as well as multiply
and divide them. You need to know how to factor!
Adding and subtracting rational expressions is just like adding and subtracting fractions
– you must have the same denominator!
EX]
Add:
5
2
 2
2 x 3x
The LCD is 6x2
=
5(3 x )
2(2)
 2
2 x(3 x ) 3 x (2)
Rewrite the fractions with LCD
=
15 x  4
15 x
4
 2 
2
6x
6x
6x2
Simplify and add numerators
SIMPLIFYING COMPLEX FRACTIONS
Simplifying complex fractions just takes patience. You have to watch your algebra as
you go through these kinds of problems.
EX]
Simplify
6
3
x 1
3
x
3( x  1)
6
6
3

x 1
x  1 ( x  1)
=
3
3
x
x
Math Analysis; updated 6/19/09
Rewrite fractions in numerator with LCD (1)
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=
3x  3
6

x  1 ( x  1)
3
x
Distribute
9  3x
x 1
=
3
x
Simplify numerator
3(3  x )
( x  1)
=
3
x
Factor numerator
(2)
(3)
(4)
=
3(3  x ) x

( x  1)
3
Multiply by reciprocal
=
x(3  x)
( x  1)
Write in simplified form
(5)
NOTES:
(6)
1) There is no need to distribute in line (6) above
2) There is more than one way to do this problem. I like to simplify
the numerator as much as possible before I begin to work with
the denominator.
Math Analysis; updated 6/19/09
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