Download Math 90, Factoring Handout

Name________________________________________
For students entering
Algebra 2 in September 2016
Summer is a time to relax and have fun, and we hope that you enjoy it! We also would like to give you the
opportunity to review and reinforce your math skills to avoid “brain drain” this summer. We have created
this preparation packet so that you focus your review in the most important areas of mathematics from the
course you completed this year to ensure success in Algebra 2. Please set aside some time each week to
keep your skills sharp by completing these problems thoughtfully. This packet will be completed and
reviewed as a portion of the initial homework assignments given in September.
LOOKING FOR HELP ? TRY THESE SUGGESTED INTERNET RESOURCES.
www.Teachertube.com
www.youtube.com
www.purplemath.com
www.youtube.com/mathdoris
www.khanacademy.org
Additionally, there are links embedded in this document that specifically address each topic. Be sure to use
your resources wisely as you strive to complete this preparation packet with accuracy. Please be sure to
bring your packet and with you to class next year.
Please SHOW ALL WORK for each problem on separate paper to receive full credit. Check your
final solutions by referring to the answer key posted on the school website. Your work will be
reviewed on the first day in second rotation that your class meets.
Please also complete the self-reflection form found at the end of this document. Please be honest
about your own mastery of the pre-requisite skills for Algebra 2.
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Part I – Arithmetic Review (You should not use a calculator to complete this section)
Order of Operations ( PEMDAS is a special acronym to help you remember)
1. Simplify expressions inside grouping symbols. (parentheses)
2. Simplify or evaluate any terms raised to powers. (exponents)
3. Do multiplication and division operations in order as you come to them from left to right. (multiply, divide)
4. Finally, do all addition and subtraction operations in order as you come to them from left to right. (add, subtract)
* Notice the words in the parentheses at the end of the lines make “PEMDAS”.
1. ) 12 ÷ 4 + 2
2.) –62 + (–6)2
3.)
6(2  5)
3(7)
4.) 12 – [3 – (4 + 32) ∙ 2]
5.) 15 - 4|6 - 10| + (-5 + 2)2
6.) 3 - |4 - 12| ∙ 4 – (8 – 7)3
7.) 12 + 5(6 + 3)2 
8.)
10.)
2
3 5 1
 
4 6 2
4
7
6
11
1 2 5
9.)   
4 5 8
3
11.)
1
2
7
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Part II – Solving Multi-Step Linear Equations
Solve each equation.
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Part III: Functions
For a detailed explanation for “functions,” go to:
http://www.mathsisfun.com/sets/function.html
1.
2.
Explain.
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3. Which of the following relations is NOT a function? Explain.
Part IV – Graphing Linear Functions
For a detailed explanation on graphing linear equations:
http://www.khanacademy.org/math/algebra/linear-equations-andinequalitie/graphing_solutions2/v/graphs-of-linear-equations
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3. Graph using the slope and y-intercept.
a)
y
2
x 3
5
b)
f ( x)  3x  6
y
y
x
x
Part V – Solving Linear Systems of Equations
For detailed explanations on solving systems on linear equations, go to:
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fast-systems-of-equations/e/systems_of_equations
Solving Systems of Equations by Graphing
By hand, graph the two lines and look for the point of intersection. Write down the point of intersection as the solution.
On the calculator, get into the form y =. Type the equations into y = for Y1 and Y2. Press 2ndTrace to get to the Calc
menu. Select 5:Intersect and press Enter, Enter, Enter. Note: the x-coordinate for the point of intersection must be in
the window for the calculator to find it.
1 Solution
No intersection=No Solution
Same Lines=Infinite Solutions
Solving Systems of Equations by Substitution
If a variable is not already alone, get ANY of the variables alone. There are always four choices (x or y in the first, x or
y in the second.), but there are usually NOT 4 good choices. Substitute into the other equation for the variable you
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solved for. Solve for the variable that is left, and then plug back into either original equation to find the other variable’s
value. It is easiest to plug into the equation with the variable already by itself.
3x  2 y  12
Example:
x  6 y  20
Step 1: First, get a variable alone. x in the 2nd equation will work well here.
3x  2 y  12
x  20  6 y
Step 2: Now substitute for the variable you solved for (x in this case) into the OTHER equation and solve.
3(20  6 y )  2 y  12
60  18 y  2 y  12
60  16 y  12
16 y  48
y3
Step 3: Plug the y-value into the equation where you solved for x.
x  20  6(3)
x2
Solution: (2,3)
Solving Systems of Equations by Elimination
Our goal is to eliminate one of the variables. To do this, we either add (when variables are opposite signs) or
subtract (when variables are the same sign.) Sometimes we need to multiply one or both of the equations to get the
coefficients (the number in front of the variable) to be the same so that the variables will cancel each other out. You
have the option of eliminating either variable every time, but there is usually one that is easier to eliminate than the
other. Solve for the variable that is left, and then plug back into either original equation to find the other variable’s
value.
 2x  y  2
 x  3 y  4
Step 1: Multiply the first equation by  3 on both sides.
3(2 x  y  2)
6 x  3y  6

 x  3 y  4
 x  3 y  4
Example:
Step 2: Now add the equations together to eliminate one of the variables (y in this case) and solve.
6 x  3 y  6
 x  3 y  4
5 x  10
x  2
Step 3: Plug the x-value into one of the original equations to find y.
 ( 2)  3 y  4
2  3 y  4
3 y  6
y  2
Solution: (2,2)
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Notice, it would have been just as easy to do this by eliminating the x-variable, in this case, by multiplying the bottom
equation by -2.
*Important Note*
If both variables drop out of the equation in either Substitution or Elimination, you must ask yourself if what you are
left with is a true statement. If yes, (for example, 2 = 2 or 0 = 0), there are infinite solutions (same line). If no, (for
example,7 = 3 or 0 = 4) there is no solution (parallel lines).
No matter which method we use, we are always trying to find the point of intersection.
Solve the system of equations by graphing. Determine whether the system has one solution, no
solution, or infinitely many solutions. If the system has one solution, name it.
1.)
y = 3x - 3
y = -½x + 4
2.)
x+y=2
-2x + y = -1
y
y
x
x
Use substitution to solve each system of equations.
3.)
x = 4y + 1
x+y=6
4.) x – 2y = 0
5x – 4y = 8
Use elimination to solve each system of equations.
5.)
-2c + b = 3
-c – b = -6
6.)
3 x + 6y = 7
2x + 4y = 22
Determine the best method to solve each system of equations. Then solve each system.
7.) -2x + 3y = 9
3x + 3y = 12
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8.) x = 2y + 6
½x – y = 3
Part VI – Factoring Polynomials
For detailed information on factoring polynomials, go to:
https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoring-quadratic-expressions/v/factoringquadratic-expressions
Factoring the Greatest Common Factor from a polynomial
Example:
1. Find the G.C.F of all terms.
24a 3b 2  4a 2 b 2  16a 2 b 4
G.C.F = 4a 2 b 2
= 4a 2 b 2 6a  1  4b 2
2. Factor the GCF from each term
of the polynomial.


Factoring by Grouping
3x  y   ax  y 
Example #1
= x  y  3  a 
Example #2
a 2b  3a 2  2b  6
=  a 2b  3a 2    2b  6 
= a  b  3  2  b  3 
2
=  b  3  a 2  2 
1. Both terms have a factor of x  y  .
2. Factor out x  y  from each term.
1. Group with parentheses the 1st two terms
and the last two terms.
2. Factor out the GCF from each group. Since
the middle sign is negative, you most also factor it out!
Notice: Both terms have a factor of b  3 .
3. Factor out b  3 from each term.
“Simple Trinomial” - Factoring a trinomial of the form x 2  bx  c (Leading coefficient is 1)
Example #1
1. What 2 numbers: MULTIPLY to = 20 and
x 2  12 x  20
ADD to = 12 ???
1  20  1  20  21

Factors of 20: 2  10  2  10  12  2 and 10!!!!
 45  4  5  9

2. List the factors of 20 and check the sums.
= x  2x  10
3. Factor.
Example #2
x 2  15x  56
1. What 2 numbers: MULTIPLY to = 56 and
ADD to = –15 ???
  1 56 
  2 28 

Factors of 56: 
  4 14 
 7  8   7    8  15
 –7 and –8!!!!
2. List the factors of –56 and check the sums.
= x  7x  8
3. Factor.
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Example #3
x 2  2 x  35
Steps 1, 2, 3 from above.
7   5  35, and 7   5  2
The numbers are 7 and –5.
= x  7x  5
“Complex Trinomial” - Factoring a trinomial of the form ax 2  bx  c , a  1 (Leading coefficient is not 1)
Factoring by Grouping (also called the “Master Product” Method, or called the “AC” method)
2x 2  7 x  4
8 1  8,
and 8   1  7
= 2 x 2  1x  8x  4
= 2 x 2  1x  8x  4
= x2 x  1  42 x  1
= 2 x  1x  4


1. Multiply 2 and –4 = –8.
2. What 2 numbers: MULTIPLY to = –8 and
ADD to = 7 ???
The numbers are 8 and –1.
3.
4.
5.
6.
Split up the 7x term  –1x + 8x
Group the 1st 2 terms and the last 2 terms.
Factor out the GCF from each group.
Factor out the common factor of 2 x  1 .
Method #2: Factoring by “Trial and Error”
2x 2  7 x  4
1, 2
1, – 4
4, –1
2, –2
–2, 2
1. Constant term is negative
2. List factors of 2 and –4.
3. Create TRIAL factors and FOIL out to pick
the CORRECT factorization.
1x  12 x  4 = 2x2  4x  2x  4  original polynomial
1x  12 x  4 = 2x 2  4x  2x  4  original polynomial
x  42 x  1 = 2x 2  1x  8x  4  2x 2  7 x  4
and so on…
= x  42 x  1 is the correct factorization.
Difference of 2 squares:
A2  B 2   A  B  A  B 
Example:
x 2  49
= x2  72
= x  7x  7
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1. Let A  x, B  7
2. Factor using the rule of difference of squares.
Factor each polynomial completely.
1. 8c3d5 - 2c2d4
2. a3 – 4a2 + 7a – 28
3. x2 – 7x + 12
4. x2 + x - 20
5. 9y2 - 64
6. 2x2 – 16x + 32
7. 4c2 + 4c + 1
8. 5x2 + 13x – 6
9. 4k2 - 100
10. 6a2 – a - 2
Part VII – Solving Quadratic Equations
For detailed information on solving quadratics by factoring, go to:
http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%201:%20Solving%20a%20quad
ratic%20equation%20by%20factoring
For detailed information on solving by completing the square, go to:
http://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/solving-quadratic-equations-by-completingthe-square
For detailed information on solving quadratics using the quadratic formula, go to:
http://www.khanacademy.org/math/algebra/quadratics/quadratic-formula/v/using-the-quadratic-formula
1. Solve each quadratic equation algebraically by factoring.
a) x2 – 7x + 12 = 0
b) x2 + x = 20
c) 9y2 = 64
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2. Solve each quadratic equation algebraically by completing the square.
a) x2 + 2x -33 = 0
b) p2 – 8p + 21 = 6
3. Solve each quadratic equation algebraically using the quadratic formula.
a) x2 + 2x – 8 = 0
b) 3x2 + 10x + 5 = 0
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Part VII – Classifying a study
Read the previous page on classifying studies and Watch the You Tube video on Experiments,
Observational studies and drawing conclusions by Jairish Patel:
https://www.youtube.com/watch?v=Z_OJzgkKe2A
Determine whether each situation describes a survey, an experiment, or an observational study. Explain
your reasoning.
A. A record label wants to test three designs for an album cover. They randomly select 50 teenagers from
local high schools to view the covers while they watch and record their reactions.
B. The city council wants to start a recycling program. They send out a questionnaire to 200 random citizens
asking what items they would recycle.
C. Scientists study the behavior of one group of dogs given a new heartworm treatment and another group
of dogs given a false treatment or placebo.
D. They yearbook committee conducts a study to determine whether students would prefer to have a print
yearbook or both print and digital yearbooks.
E. A pharmaceutical company wants to test whether a new medicine is effective.
F. A news organization wants to randomly call citizens to gauge opinions on a presidential election.
G. A research company wants to study smokers and nonsmokers to determine whether 10 years of smoking
effects lung capacity.
H. A national pet chain wants to know whether customers would pay a small annual fee to participate in a
rewards program. They randomly select 200 customers and send them a questionnaire.
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Summer Packet Self-Reflection Sheet
Name: __________________________________
Date: __________
Math course completed 2015-2016:
Incoming grade: 9 10 11
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Answer each group of questions in a short paragraph. Feel free to extend your comments onto another
sheet of paper if necessary. Be specific!
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1.
Please describe the window of time that you used to complete this review assignment.
2.
Use the space below to list the topics that you found to be most comfortable.
3.
Use the space below to list the topics that you found to be least comfortable.
4.
What will you do to remediate the areas that you found to be least comfortable?