Download Summer Mathematics Packet

Sail into Summer with Math!
For Students Entering Algebra II
The Pyle Math Department requests that students review math skills over the summer. Reviewing
core concepts helps students retain math knowledge and helps them prepare for success upon return
to school in the fall.
Students may use the online program skills tutor OR this review packet. EITHER the completed onlineskills tutor lessons OR the summer review packet is due to the student’s math teacher on the first day
of school.
Students are welcome to try other course assignments, as either review or enrichment. However,
completion of the next level’s assignment is not a pathway into the next course.
Students who are accelerating through a math course are expected to complete the lessons for both
the course that is passed over AND the course in which the student is enrolled in for the upcoming
school year. For example, a student who just completed Math 5 in 5th grade and is registered for
Math 7 in 6th grade should complete the lessons for both Math 6 and Math 7. This is regardless of
teacher recommendation or a pre-approved parent placement change request.
Say Goodbye to Geometry and Hello to Algebra 2!
Summer Mathematics Packet
Table of Contents
Page
1
Objective
Order of Operations
Suggested Completion Date
June 25
2
Solving Equations
June 29
3
Exponents
July 3
4
Addition and Subtraction of Polynomials
July 7
5
Multiplying Polynomials
July 11
6
Division of Polynomials
July 15
7
Factor Polynomials I
July 19
8
Factor Polynomials II
July 23
9
Factor Polynomials III
July 27
10
Solving Systems of Equations by Substitution
July 31
11
Solving Systems of Equations by Elimination
August 4
12
Simplifying Radicals
August 8
13
Rationalizing Radicals
August 12
14
Solving Radical Equations
August 16
15
Quadratic Formula
August 20
Summer
90-1-1
Summer Mathematics Packet
Order of Operations
Hints/Guide:
The rules form multiplying integers are:
positive x positive positive
positive x negative = negative
negative x negative positive
negative x positive negative
=
=
=
The rules for dividing integers are the same as for multiplying integers.
(PEMDAS)
REMEMBER: Order of Operations
P- parenthesis
E - exponents
M/D - multiply/divide which comes first
A/S- add/subtract which comes first
Exercises: Solve the following problems. Show all work.
1.
100- 15
9+8
2.
3 + 4[ 13 - 2(6 - 3)}
3.
5[2(8 + 5) - 15}
4.
14 +6 2 - 8 .;. 4
5.
7(14) - 3(6)
2
6.
14.;. [3(8 - 2) - 11]
7.
32"'-(-7+ 5)3
8.
•
Use grouping symbols to make each equation true.
9.
6+8.;.4·2=7
Algebra
2
10.
-1-
5+ 4 3 - 1
•
=
18
Summer
2,*1
Summer Mathematics Packet
Solving Equations
Hint/Guide:
Equation-Solving
1.
Procedure
Multiply on both sides to clear the equation of fractions or decimals.
2.
3.
4.
5.
Distribute.
Collect like terms on each side,
if necessary.
Get all terms with variables on one side and aU constant terms on the other side, using
the addition principle.
Collect like terms again,
if necessary.
6.
Multiply or divide to solve for the variable, using the multiplication principle.
7.
Check all possible solutions in the original equation.
-6 6
6 - � �16(r )
O
£x ! + !x = Z + 2x
3
2
Example:
x
Multiply by LCM of
2X
+
6
4x - 1 + 3x = 7 + 12x
Simplify.
7x
Combine like terms.
1
=
7 + 12x
7x- 12x = 7 + 1
- 5x = 8
x = -!l,
5
both sides.
Collect variables on one side and constants on other.
Combine like terms.
Divide by -5.
Exercises: Solve each equation. Show all work.
1.
3(r - 6)+ 2
3.
1 (6x+ 24) - 20
3
5.
0.7(3x+ 6)
Algebra
2
=
=
4(r+ 2) - 21
=
-
1 (12x - 72)
2.
5(t+ 3)+ 9
4.
1: (3x+ 4)
5
6.
a+ (a - 3)
4
1.1 - (x+ 2)
-2-
=
3(t - 2)+ 6
=
20
=
(a+ 2)- (a+ 1)
Summer
b( 0 J
I
Summer Mathematics Packet
Exponents
Hints/Guide:
Rules for Exponents
Negative Exponents:
aO =
I
a-n = 1
an
when a to °
when a toO
am = am
Product Rule:
Quotient Rule:
Power Rule:
Product 10 Power:
-
n
an
(ab)n = a"bn
Quotient to a Power:
Exercises: Simplify using the Rules for Exponents.
2.
3.
(4a)3. (4a)B
5.
8.
11.
13. Express using a positive exponent:
14. Express using a negative exponent:
1
I
Algebra
2
-3-
Summer
:;2. 0 II
Summer Mathematics Packet
Addition and Subtraction Polynomials
Hint/Guide:
Only like terms can be added or subtracted.
Like terms have the same variables with the same exponents.
Only the coefficients (numbers) are added or subtracted.
A subtraction sign in front of the parenthesis changes each term in the parenthesis to the
opposite.
•
•
•
•
Examples:
1) Add the polynomial.
2) Subtract the polynomial.
(3..1 - 2x + 2) + (5K - 2..1 + 3x - 4)
= 5K + (3 - 2)..1 + (-2 + 3)x+(2
= 5K + x' + x- 2
-
4)
(9K + K - 2..1 + 4) - (2K+x' - 4K - 3..1)
= 9K + K - 2..1 + 4
2K - x· + 4K + 3..1
= 7K - x4 + 5K + x'+4
-
Exercises: Add or subtract the polynomials. Show all work.
1.
(3x+ 2)+ (-4x+ 3)
2.
(-6x+ 2)+ (x' + x-3)
3.
(1.2K + 4.5x' - 3.8x)+
(-3.4K - 4.7x' + 23)
4.
(114 X4 + 2/3K + 5/8x' + 7)+
(-3/4x 4 + 3/8x2- 7)
5.
(6x+ 1)- (-7x+ 2)
6.
(x' - 5x+ 4)- (8x- 9)
7.
(0.5x4 - 0.6x' + 0.7) (2.3x4 + 1.8x- 3.9)
8.
(1I5K+ 2x'- 0.1)­
(-2/5K + 2x' + 0.01)
Algebra
2
- 4-
Summer
1. D \ \
SUlTlmer Mathematics Packet
Multiplying Polynomials
Hint/Guide:
Multiply the coefficients and use the rule of exponents for the variables.
Remember: FOIL F - first 0 - outers I inners L - last
OR Box Method
•
•
-
Examples:
1. ()' + 2x - 3){)' + 4)
=)' )' + 2x· )' - 3 . )' + )' 4+ 2x·4 - 3 . 4
= ;/' + 2x' - 3)' + 4)' + 8x- 12
2. (x + 5)(x + 4)
•
•
= x' + 2x' + K+ 8x- 12
x
x
4
5
�
8illQJ
= K+ 9x+ 20
Exercises: Multiply the polynomials. Show all work.
2.
-4x(2K - 6>1 - 5x+ 1)
1.
-3x(x - 1)
4.
(5 - x)(5- 2x)
6.
(K+ >I + x+ l)(x- 1)
7.
9.
(3x+ 2)(4)1+ 5)
10.
Algebra·2.
5.
3.
(x+ 5)(x- 2)
(3d - 5a+ 2)(2£1 - 3a+ 4)
8.
- 5-
(1/5>1+ 9)(3/5>1- 7)
SUlTlmer
OJ 6 I
I
Summer Mathematics Packet
Division of Polynomials
Hint/Guide:
Divide the coefficients.
Use the rules of exponents to divide the variables.
•
•
Example:
x' +lOx'+ax
2x
=
=
=
x' + lax' + ax
2x
2x
2x
! x'.1 +10 x'.1 +§. x·1
2
2
2
l x' + 5x+4
2
Exercises: Divide the polynomials. Show all work.
1.
24x4 - 4K + Jt -16
8
4.
(25f
6.
(18X' - 27)i
Algebra
2
+
u-2J - if
u
2.
151 - 30t) -'" (5t)
-
3K) -'" (9K)
·6·
3.
x
5.
( 24X' + 32)i 8Jt).;. (-8Jt)
7.
91-1 + 31-s- 6r1
3rs
-
Summer
9 bIt
Summer Mathematics Packet
Factor Polynomials I
Hint/Guide:
Always look for a common factor first. Don't forget to include the variable in the
common factor.
Check your answer by multiplying.
•
•
Example:
Factor
15K - 12x4 + 27x' - 3x'
Question: What number is common to the coefficients of 15, 12,27, and 3?
Answer: 3
Question: What exponent is common to variables of K, X4, x',and x'?
Answer: x'
=
(3x')(5x'l) - (3x')(4x') + (3x')(9x) - (3x')(1)
=
3x'(5x' - 4x' + 9x - 1)
Exercises: Factor the polynomials. Show all work,
1.
>1 - 4x
4.
6>1 + 3x- 15
5.
6.
16X'/ - 32)i1 - 48.-0/
7.
8.
1.6x4 - 2.4�
2.
+
3.
3.2>1 + 6.4x
9.
-7-
17)i1
5/3X'
+
+
34�1
4/3X'
+
+
51xy
1/3x4
+
1/3�
summer;
a. 6 /1
Summer Mathematics Packet
Factor Polynomials n
Hints/Guide:
Write the terms in descending order.
List the factors for the constant term.
Add those factors to find the match for the coefficient of the middle term.
•
•
•
Example:
Factor f-
-
24 + 5t.
Write in descending order:
List the factors:
The factors are:
f- + 5t- 24
Pairs of Factors Sums of Factors
-1,24
23
-2,12
10
-3, 8
5
-4,6
2
(t- 3 )(t+ 8)
Exercises: Factor the polynomials. Show all work.
1.
;/ + 5x + 6
2.
1 + Hy + 28
3.
;/-8x+ 15
4.
;/ + 2x-15
5.
-2x-99 + ;/
6.
;/ -72 + 6x
7.
a4 + 2J -35
9.
;/ + 20x + 100
10.
;/ -25x + 144
12.
d + 5mn + 4rf
Algebra 2
11.
J-2ab-3d
-8-
Sum""""
'L \J \ \
Sumrrer Mathematics Packet
Factor Polynomials III
Hint/Guide:
To Factor Polynomials of the type ax2 + bx+ c, when at 1:
Write the terms in descending order.
Factor all common factors.
List the factors of the coefficient of the first term,
List the factors for the constant term.
Multiply a factor of 1st term with 3rd term. Multiply the other factor of 1st
term with 3rd term. Add the two together, to find middle term. Continue this
process until the correct factor combination is found.
•
•
•
•
•
Example:
Factor 24x" - 76x+ 40.
4(6x2 - 19x + to)
Factor the common factor:
Find the factors of the first and third
Factors of 1sf Term
terms:
1,6 OR -1, -6
2,3 OR -2, -3
Try combinations of factors:
The factors are:
a)
b)
c)
d)
3, -1 + to ' 2 = -23
3 ' -to + -1 . -2 = -32
3, -2 + -5 ' 2 = -16
3 . -5 + -2 2 = -19
•
Factors of 3rd Term Middle Term
l,tO OR -1, -to
2,5 OR -2, -5
wro"9
wro"9
wro"9
correct
4(3x - 2)(2x -5)
Exercises: Factors the polynomials. Show all work.
1.
2K-7x-4
2.
3K- 4x-15
3.
3K+ 4x+1
4.
9K+6x- 8
5.
6.
18K -3x-10
7.
14K+19x-3
8.
9.
24K + 47x-2
10.
18K -21K-9x
11.
12.
15£1-5ab-20d
Algebro
2
9K+18x-16
-9-
Sumrrer
'LD I
I
Sum"",r
Mathematics Packet
Solving Systems of Equations by Substitution
Hint/Guide:
•
•
•
•
•
Example:
Solve one equation for a variable with a coefficient of 1.
Substitute what the variable equals into the other equation of the original
pair. (The new equation should have only one variable.)
Solve for the variable.
Use that answer to solve for the other variable.
Answers are ordered pairs: (x, y)
Solve
x- 2y= 6
3x + 2y=4.
x =6 + 2y
Solve the first equation for x:
Substitute your answer above into the second equation:
Distribute:
Combine like terms:
Collect like terms to one side (subtract 18 both sides):
Isolate the variable (divide by 8 both Sides):
3(6 + 2,n + 2y =4
18 + 6y + 2y=4
18 + 8y= 4
8y =-14
y= -14 OR -7
8
4
Substitute the y value into an original equation to solve for x:
x- 2(-7/4) =6
x - (-14/4) = 6
X = 10/4 or 5/2
The solution to the system of equations: (5/2. -7/4)
Exercises: Solve the system of equations using the substitution method. Show all work.
1.
s+t= 4
s-t= 2
2.
x-y= 6
x+y= -2
3.
y- 2x=-6
2y-x= 5
4.
x-y= 5
x+2y= 7
5.
2x+3y=-2
2x-y= 9
6.
x+ 2y= 10
3x+.4y= 8
-
Algebra 2
- 10-
Sum"",r
1 q\ \
Summer Mathematics Packet
Solving Systems of Equations by Elimination
Hint/Guide:
•
•
•
Answers are ordered pairs (x, y).
Eliminate one variable by adding the two equations together.
Sometimes,one equation must be multiplied by a number to have a variable with the same coefficient.
Examples:
1.
Solve 2x+ 3y=8
x+ 3y=7
Multiply the equation by -1 to make the ycoefficients opposite:
Add the equations together:
Solve for yby substituting the value of xinto the original equation:
Solve the equation for y.
2x + 3 y= 8
-x - 3 y= -7
x+ Oy= 1
x= 1
2(1) + 3 y= 8
3 y=6
y= 2.
The solution for this system: (1,2)
2. Solve
3x + 6y =-6
5x - 2y =14
Multiply the second equation by 3 to get the ycoefficients the same:
Add the equations together:
Solve for x.
Solve for yby substitutios the value of x into the original equation:
Solve the equation for y.
The solution for this system: (2, -2)
3 x+ 6y=-6
15x - 6y= 42
18x+ Oy=36
x= 2
3(2) + 6y=-6
6y= -12
Y = -2
Exercises: Solve the systems of equations by elimination. Show all work.
2,
x-y= 7
3.
3x-y= 9
4.
4x-y = 1
1.
x+ y=10
x+y = 3
2x+ y = 6
3x+ y= 13
x- y= 8
5.
-x-y = 8
2x-y = -1
Algebro
2
6.
7.
3x-y = 8
x+2y= 5
- 11 -
2w- 3z= -1
3w+4z= 24
8.
3x-4y=16
5x+6y= 14
Summer
2 () I[
Summer Mathematics Packet
Simplifying Radicals
Hint/Guide:
For any real number that is not negative, [(xj
•
•
Assume that the radical sign (./) extends
parenthesis.
Examples:
1.
Simplify 1(34
=
2. Simplify I(a-If) =
=
over
x
the entire expression in
3x
Db
3. Simplify 1(>1 + 2x + 1)
=
I(x + l)(x + 1) =
4. Simplify by factoring: 1(32x15)
=
x
+ 1
1[(16 ' 2)(X14 x)]
•
=
4K[(2�
Exercises: Simplify the radical expressions.
2.
1.
5.
[(34d/
9.
[(4J1-20x+25)
12.
[(20)
16.
[(8J1+8x+2)
Algebra 2
[(9J1)
6.
13.
10.
1(48x)
17.
3.
4.
7.
8.
[(9,1 +12p+4)
14.
1(64/)
[(27J1-36x+12)
- 12 -
[(J -10a+ 25)
11.
15.
[(75)
[(20Jl)
18.
[(x-2J1 + K)
Summer
&b \\
•
Summer Mathematics Packet
•
Rationalizing Radicals
Hint/Guide:
Assume that the radical extends over the entire expression in parenthesis.
To rationalize a radical expression, first simplify through division if possible.
Multiply the numerator and the denominator by the denominator.
Simplify the expression.
•
•
•
•
Examples:
1.
Rationalize the radical expression: 1(30';)
1(6d)
2. Rationalize the radical expression: 1(3)
1(2)
3. Rationalize the radical expression: 1(5)
I(x)
=
=
1(5d) = I(d . 5a) = I(d) . 1(5a) = al(5a)
1(3) .1(2)
1(2) 1(2)
=
=
/(6) = 1(6)
1(4)
2
1(5) . I(x) = 1(5x)
x
I(x) I(x)
Exercises: Rationalize the radical expressions. Show all work.
1.
3.
2.
[(18)
[(60)
[(15)
[(2)
ill)
[(75)
4.
/(18)
[(32)
5.
[(18b)
[(2b)
6.
[(48X»
[(3x)
7.
[(2)
[(5)
8.
2
[(2)
9.
[(3)
[(x)
10.
[(27 c)
[(32c)
11.
[(V)
[(xy)
12.
[C16a4b6)
[(128cfb6)
Algebra 2
. 13 .
Summer
ao
1/
-Sumrrer Mathematics Packet
Solving Radical Equations
Hint/Guide:
•
•
•
•
•
Assume that the radical extends o>er the entire expression in parenthesis.
Isolate the radical.
Square both sides to remove the radicals.
Combine like terms.
Isolate the IIClriable to one side.
Examples:
2l(x+ 2) = f(x+ 10)
1. Solve
Square both sides:
Distribute:
Collect like terms:
Divide to isolate the IIClriable:
x - 5 = f(x+ 7)
2. Solve
Square both sides:
(21(x+ 2))2 = (f(x+ 10))2
4(x+ 2) = X + 10
4x + 8 = x + 10
4x - x =10 - 8
3x=2
x=2/3
(x- 5)2 = (f(x+ 7))2
,,; - lOx + 25 = x+ 7
,,; - llx + 18 = 0
(x- 9)(x- 2) = 0
x-9 = O
x- 2 = O
x=9
OR x=2
Combine like terms:
Factor:
Solve
Exercises: Solve the radical equations. Show all work.
2.
[(x) 7
1.
[(x) 5
=
=
4.
[(x+ 4)
7.
[(5x- 7)
10.
x- 7
Algebra 2
=
=
11
=
[(x+ 10)
[(x- 5)
5.
3+ [(x- 1)
B.
[(4x- 5)
11.
x- 9
=
- 14-
=
=
5
[(x+ 9)
[(x- 3)
3.
[(x+ 3) = 20
6.
4+ [(y- 3) = 11
9. [(2y+ 6)
=
12. U(x- 1)
Sumrrer :
[(2y- 5)
=
x- 1
cl b \ \
Summer Mathematics Packet
Quadratic Formula
Hint/Guide:
Assume that the radical extends over the expression in parenthesis.
Equation must be in the form ax2 + bx + c= 0 (standard form).
Try to factor first.
If you cannot find factors, then use the quadratic equation.
•
•
•
•
Quadratic
x =
Example:
-b
±
Formula
,j(b2
2a
-
4acl
Solve Jt = 4x+7
Jt - 4x - 7 = 0
a = 1. b = -4. c = -7
x = (-4):t [[(_4) 2 - 4(1)(-7)J
2(1)
x = 4 :t [(16 +28)
2
Write the equation in standard form:
Identify a. b. and c for the formula:
Substitute into the formula:
-
Simplify:
x =
x =
Simplify the radical:
x =
Solutions:
x =
2
+
J(ll) or
x =
4 ;[(4 4)
2
4+21(11)
2
2 ±f(II)
2 - J(ll)
Exercises: Solve using the quadratic formula. Show all work.
2.
,>1 = 6x-9
1.
,>I-4x= 21
3.
3/-7y+4=0
4.
,>1-9=0
5.
,>I-2x-2= 0
6.
,>I-4x-7=0
7.
l-lOy+22 = 0
8.
1 + 6y= 1
9.
2/-6y=10
Algebra 2
- 15 -
Summer,;). 0 t
\