For Students Entering Algebra II - Montgomery County Public...

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 online-skills 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!

Page 1 2 3

Summer Mathematics Packet

Table of Contents

Objective Order of Operations

Solving Equations

Exponents

Suggested Completion Date June 25 June 29 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

Hints/Guide:


Order of Operations

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.

REMEMBER: Order of Operations (PEMDAS)

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.

3.

5.

7.

100 - 15 9+8

5[2(8 + 5) - 15}

7(14) - 3(6) 2

32"'-(-7+ 5)3

2. 3 + 4[ 13 - 2(6 - 3)}

4. 14 + 6 • 2 - 8 .;. 4

6. 14.;. [3(8 - 2) - 11]

8.

Use grouping symbols to make each equation true.

9. 6+8.;.4·2=7

Algebra 2 - 1 -

10. 5 + 4 • 3 - 1 = 18

Summer 2,*1


Solving Equations

Hint/Guide:

Equation-Solving Procedure 1. Multiply on both sides to clear the equation of fractions or decimals. 2. Distribute. 3. Collect like terms on each side, if necessary. 4. Get all terms with variables on one side and aU constant terms on the other side, using

the addition principle. 5. 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.

Example: £x - ! + !x = Z + 2x 3 6 2 6 6 Ox -� + �16(r 2X)

4x - 1 + 3x = 7 + 12x

7x - 1 = 7 + 12x

7x - 12x = 7 + 1 - 5x = 8

x = -!l,

Multiply by LCM of 6 both sides.

Simplify.

Combine like terms.

Collect variables on one side and constants on other. Combine like terms.

5 Divide by -5. Exercises: Solve each equation. Show all work.

1. 3(r - 6) + 2 = 4(r + 2) - 21

3. 1 (6x+ 24) - 20 = - 1 (12x - 72)

3 4

5. 0.7(3x+ 6) = 1.1 - (x+ 2)

Algebra 2

2.

4.

6.

- 2 -

5(t+ 3) + 9 = 3(t - 2) + 6

1: (3x+ 4) = 20

5

a+ (a - 3) = (a+ 2) - (a+ 1)

Summer b( 0 J I


Exponents Hints/Guide:

Rules for Exponents

Negative Exponents:

Product Rule:

Power Rule: Quotient to a Power:

a-n = 1 an

aO= I

Exercises: Simplify using the Rules for Exponents.

2.

5.

8.

11.

13. Express using a positive exponent:

Algebra 2 - 3 -

when a to °

when a toO

Quotient Rule: am = am - n

an

Product 10 Power: (ab)n = a"bn

3. (4a)3. (4a)B

14. Express using a negative exponent:

1 I

Summer :;2. 0 II


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 - 4) = 5K + x' + x- 2

(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.

3.

5.

7.

(3x+ 2) + (-4x+ 3)

(1.2K + 4.5x' - 3.8x) + (-3.4K - 4.7x' + 23)

(6x+ 1) - (-7x+ 2)

(0.5x4 - 0.6x' + 0.7) -

(2.3x4 + 1.8x- 3.9)

Algebra 2

2.

4.

6.

8.

- 4-

(-6x+ 2) + (x' + x-3)

(114 X4 + 2/3K + 5/8x' + 7) +

(-3/4x4 + 3/8x2 - 7)

(x' - 5x+ 4) - (8x- 9)

(1I5K + 2x' - 0.1)

(-2/5K + 2x' + 0.01)

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

= x' + 2x' + K + 8x- 12

2. (x + 5)(x + 4)

x 4

x 5

� 8illQJ

= K+ 9x+ 20

Exercises: Multiply the polynomials. Show all work.

1. -3x(x - 1) 2.

4. (5 - x)(5 - 2x)

6. (K + >I + x+ l)(x- 1)

9. (3x+ 2)(4)1 + 5)

Algebra·2.

-4x(2K - 6>1 - 5x+ 1) 3. (x+ 5)(x- 2)

5. (3d - 5a+ 2)(2£1 - 3a+ 4)

7. 8.

10. (1/5>1 + 9)(3/5>1- 7)

- 5 - SUlTlmer OJ 6 I I


Division of Polynomials Hint/Guide:

• Divide the coefficients. • Use the rules of exponents to divide the variables.

Example: x' + lOx' + ax = x' + lax' + ax

2x 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

2. u-2J - if u

3. x

4. (25f + 151 - 30t) -'" (5t) 5. (24X' + 32)i - 8Jt).;. (-8Jt)

6. (18X' - 27)i - 3K) -'" (9K)

Algebra 2 ·6·

7. 91-1 + 31-s- 6r1 3rs

Summer 9 bIt


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 + 27 x' - 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 2. 3.

4. 6>1 + 3x- 15 5. 17)i1 + 34�1 + 51xy

6. 16X'/ - 32)i1 - 48.-0/ 7.

8. 1.6x4 - 2.4� + 3.2>1 + 6.4x 9. 5/3X' + 4/3X' + 1/3x4 + 1/3�

- 7 - summer;a. 6 /1

Hints/Guide:


Factor Polynomials n

• 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: f- + 5t- 24

Pairs of Factors List the factors: -1,24

-2,12 -3, 8 -4,6

The factors are: (t- 3)(t+ 8)

Sums of Factors 23 10 5 2

Exercises: Factor the polynomials. Show all work.

1. ;/ + 5x + 6

4. ;/ + 2x-15

7 . a4 + 2J -35

10. ;/ -25x + 144

Algebra 2

2. 1 + Hy + 28

5. -2x-99 + ;/

11. J -2ab-3d

- 8 -

3. ;/-8x+ 15

6. ;/ -72 + 6x

9. ;/ + 20x + 100

12. d + 5mn + 4rf

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. Factor the common factor: 4(6x2 - 19x + to) Find the factors of the first and third terms: Factors of 1 sf Term

1,6 OR -1, -6 2,3 OR -2, -3

Try combinations of factors: a) 3, -1 + to ' 2 = -23 b) 3 ' -to + -1 . -2 = -32 c) 3, -2 + -5 ' 2 = -16 d) 3 . -5 + -2 • 2 = -19

The factors are: 4(3x - 2)(2x -5)

Exercises: Factors the polynomials. Show all work. 1. 2K-7x-4 2. 3K- 4x-15

wro"9 wro"9 wro"9 correct

4. 9K + 6x- 8 5. 9K + 18x-16

7. 14K + 19x-3

10. 18K -21K -9x

Algebro 2

8.

11.

- 9 -

Factors of 3rd Term Middle Term l,tO 2,5

OR -1, -to OR -2, -5

3. 3K + 4x+ 1

6. 18K -3x-10

9. 24K + 47x-2

12. 15£1 -5ab -20d

Sumrrer 'LD I I

Sum"",r Mathematics Packet

Solving Systems of Equations by Substitution Hint/Guide:

• 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)

Example: Solve x- 2y= 6 3x + 2y= 4.

Solve the first equation for x: x= 6 + 2y 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.

4.

s+ t= -4 s-t= 2

x-y= 5 x+ 2y= 7

Algebra 2

2.

5.

x-y= 6 x+ y= -2

2x+ 3y= -2 2x-y= 9

- 10-

3.

6.

y- 2x= -6 2y-x= 5

x+ 2y= 10 3x+.4y= 8

Sum"",r 1 q \ \


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.

The solution for this system: (1,2)

2. Solve 3x + 6y = -6 5x - 2y = 14

Multiply the second equation by 3 to get the y coefficients the same: Add the equations together: Solve for x.

Solve for y by substitutios the value of x into the original equation: Solve the equation for y.

The solution for this system: (2, -2)

2x + 3 y= 8 -x - 3 y= -7 x+ Oy= 1

x = 1

2(1) + 3y= 8 3y= 6

y= 2.

3x+ 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.

1. x+ y= 10 2, x-y= 7 3. 3x-y= 9 4.

5.

x- y= 8 x+ y = 3 2x+ y = 6

-x-y = 8 2x-y = -1

Algebro 2

6. 3x-y = 8 x+ 2y= 5

- 11 -

7. 2w- 3z= -1 3w+ 4z= 24

8.

4x-y = 1 3x+ y= 13

3x-4y= 16 5x+ 6y= 14

Summer 2 () I[


Simplifying Radicals Hint/Guide:

• For any real number that is not negative, [(xj = x • Assume that the radical sign (./) extends over the entire expression in

parenthesis.

Examples: 1. Simplify 1(34 = 3x

2. Simplify I(a-If) = Db

3. Simplify 1(>1 + 2x + 1) = I(x + l)(x + 1) = x + 1

4. Simplify by factoring: 1(32x15) = 1[(16 ' 2)(X14 • x)] = 4K [(2�

Exercises: Simplify the radical expressions.

1. 2. [(9J1) 3.

5. [(34d/ 6. 7.

9. [(4J1-20x+ 25) 10. [(9,1 +12p+ 4)

12. [(20) 13. 1(48x) 14. 1(64/)

16. [(8J1 + 8x+ 2)

Algebra 2

17. [(27J1-36x+ 12)

- 12 -

4.

8. [(J -10a+ 25)

11. [(75)

15. [(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) = /(6) = 1(6) 1(2) 1(2) 1(4) 2

= 1(5) . I(x) = 1(5x) I(x) I(x) x

Exercises: Rationalize the radical expressions. Show all work. 1. [(18) 2. [(60) 3. ill)

[(2) [(15) [(75)

4. /(18) [(32)

7. [(2) [(5)

10. [(27 c) [(32c)

Algebra 2

5. [(18b) [(2b)

8. 2 [(2)

11. [(V) [(xy)

. 13 .

6. [(48X» [(3x)

9. [(3) [(x)

12. [C16a4b6) [(128cf b6)

Summer ao 1/

Hint/Guide:

-Sumrrer Mathematics Packet

Solving Radical Equations

• 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: 1. Solve 2l(x+ 2) = f(x+ 10)

Square both sides:

Distribute: Collect like terms:

Divide to isolate the IIClriable:

2. Solve x- 5 = f(x+ 7) Square both sides:

Combine like terms: Factor: Solve

(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

Exercises: Solve the radical equations. Show all work.

1. [(x) = 5 2. [(x) = 7

4. [(x+ 4) = 11

7. [(5x- 7) = [(x+ 10)

10. x- 7 = [(x- 5)

Algebra 2

5. 3 + [(x- 1) = 5

B. [(4x- 5) = [(x+ 9)

11. x - 9 = [(x- 3)

- 14-

3. [(x+ 3) = 20

6. 4 + [(y- 3) = 11

9. [(2y+ 6) = [(2y- 5)

12. U(x- 1) = x- 1

Sumrrer : cl b \ \


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 Formula

x = - b ± ,j(b2 - 4acl 2a

Example: Solve Jt = 4x+ 7

Write the equation in standard form: Identify a. b. and c for the formula: Substitute into the formula:

Simplify:

Simplify the radical:

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

x = 4 ;[(44) 2

x = 4 + 21(11) 2

x = 2 ±f(II)

Solutions: x = 2 + J(ll) or x = 2 - J(ll)

Exercises: Solve using the quadratic formula. Show all work.

1. ,>I -4x= 21 2. ,>1 = 6x-9 3.

4. ,>1-9=0 5. ,>I -2x-2 = 0 6.

7. l-lOy+ 22 = 0 8. 1 + 6y= 1 9.

Algebra 2 - 15 -

3/-7y+4=0

,>I-4x-7=0

2/-6y= 10

Summer,;). 0 t \

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