algebra 2 summer homework packet - sound...

Algebra 2 Summer

Homework Packet I know, you will probably groan when you see this but it will help set you up for success in Algebra 2. You will get a thorough fast paced review of Algebra 1 when you start the term so consider this your warm up lap before the sprint starts.

I cannot emphasize enough how important it is to be NEAT when working problems, to be very NEAT when working problems, to be exceedingly NEAT when working problems. Your problems are going to get complicated in a hurry and if you can't discipline yourself to be neat with short simple ones, you certainly won't be able to keep it all organized with big problems.

You must get in the habit of showing every step, as if you were explaining this to a 6̂ ^ grader. Show me every step. I f you find an error, a simple cross-out will do, then start on a fresh line. Erasing is never required.

Here are some helpful websites you may find useful if you get stuck on your summer packet:

www.nmthforum.orq/library/drmath/drmath.middle.html www.purplemath.com www.virtualnercl.com

www.math.com www.freemathhelp.com/alRebra-help

Cheers,

Mrs. Kowalski

A. Simplifying Polynomial Expressions

1. Combining L i k e Terms

- You can add or subtract terms that are considered "Hlce", or terms that have the same variable(s) wi th the same exponent(s).

Ex. 1: 5 x - 7 y + 10x + 3y

5x ; ; Jy + lOx + l y

1 5 x - 4 y

Ex.2: - 8 h ^ + l O h ' - 12h^-15h^

-8h^+ lOh^- I2i r^- 15h^

-20h^ - 5h^

I L Applying the Distributive Property

Every tenn inside the parentheses is multiphed by the term outside o f the parentheses.

^-x. 1: 3 ( 9 x - 4 ) Ex.2: Ax^{5x^+6x)

3 • 9A: - 3 • 4 4 x ' • 5x ' + 4x-

2 7 X - 1 2 2 0 x ' + 2 4 x - '

I I L Combining L ike Terms A N D the Distributive Property (Problems with a Mix!)

- Sometimes problems w i l l require you to distribute A N D combine like terms!!

£ x . l : 3 ( 4 x - 2 ) + 13x £x . 2 : 3(12x - 5) - 9 ( -7 + lOx)

3 - 4 x - 3 - 2 + 1 3 x 3 - 1 2 x - 3 - 5 - 9 ( - 7 ) - 9 ( 1 0 x )

1 2 x - 6 + 13x 3 6 x - 1 5 + 6 3 - 9 0 x

2 5 x - 6 - 5 4 X + 48

I

P R A C T I C E S E T 1

Simplify.

1. 8x-9y + \6x + \2y

3. 5n-{3-4n)

5. ]Qq(\6x + \\)

7. 3 (18z -4w)+2(10z -6w)

9. 9 ( 6 x - 2 ) - 3 ( 9 x ^ - 3 )

2. 14v + 22-15v^ +23.V

4. -2(11/) -3)

6. - ( 5 x - 6 )

8. (8c + 3 ) + i 2 ( 4 c ' - i 0 )

10. - ( v - x ) + 6(5x + 7)

2 .

B. Solving Equations

I . Solving Two-Step Equations

A couple o f hints: 1. To solve an equation, U N D O the order of operations and work in the reverse order.

2. R E M E M B E R ! Addit ion is "undone" by subtraction, and vice versa. Multipl ication is "undone" by division, and vice versa.

£x . 1: 4 x - 2 = 30 £x . 2 : 87 = - 1 LY + 21

+ 2 + 2 - 2 1 - 2 1

4x = 32 66 = - l l x

^ 4 + 4 ^ - 1 1 ^ - 1 1

X = 8 - 6 = X

I I . Solving Multi-step Equations With Variables on Both Sides of the Equal Sign

When solving equations wi th variables on both sides o f the equal sign, be sure to get all terms wi th variables on one side and all the terms without variables on the other side.

Ex. 3; 8x + 4 = 4x + 28

- 4 - 4

8x = 4 x + 24

- 4 x - 4 x

4x = 24

^ 4 + 4

x = 6

I I I . Solving Equations that need to be simplified first

In some equations, you w i l l need to combine like terms and/or use the distributive property to simplify each side o f the equation, and then begin to solve h.

^ x . 4 ; 5 ( 4 x - 7 ) = 8x + 45 + 2x

2 0 x - 3 5 = 10x + 45

- l O x - l O x

1 0 x - 3 5 = 45

+ 35 +35

10x = 80

+10 +10

x = 8

3


Solve each equation. You must show all work.

1. 5 x - 2 = 33 2. 140 = 4x + 36

3. 8 ( 3 x - 4 ) = 196 4. 45.x-- 720 + 15x = 60

5. l32 = 4(12x-9) 6. 198 = 154 + 7A - -68

7. -131 = -5(3A - -8 ) + 6X i. - 7A--10 = 18+ 3.Y

9. 12x + 8 - l 5 = - 2 ( 3 x - 8 2 ) 10. - ( 1 2 x - 6 ) = i2x + 6

I V . Solving Literal Equations

- A literal equation is an equation that contains more than one variable. - You can solve a literal equation for one o f the variables by getting that variable by itself

(isolating the specified variable).

Ex. 1 : 3xv = 18, Solve for x.

3xy 18

3 V " 3y

_6_

V

Ex. 2: 5a-\0b = 20, Solve for a.

+ 10/) =+ 10/?

5a = 20+106

5a 20 10/7 — = — +

5 5 5 a = A + 2b


Solve each equation for the specified variable.

1. Y+V=W, for K 2. 9 w r = 8 1 , f o r w

3. 2 ^ - 3 / = 9, f o r / 4. clx + t= 10, for X

5. P = ( g - 9 ) 1 8 0 , f o r g 6. 4.r + ;̂ - 5/2 = 1 Oy + u, for x

C. Rules of Exponents

Multipl icat ion: Recall (x'")ix") = x^"'^"' £ 'x:(3x V ) ( 4 x / ) = ( 3 • 4 ) ( x ' • x ' ) ( v " • / ) = I2A--V"

Division: Recall — = x""-"' x"

Ex: 42m\r - 3m\/

I 42 \

in [ J = - \4rir J

Powers: Recall (x'")" = x'"'"' Ex: i-la'hc'y = {-2y(a'f{h'f(c'f = -Sa'b'c''

Power o f Zero: Recall x" = \, x ^ 0 Ex: 5x%' = ( 5 ) ( 1 ) ( / ) = 5


Simplify each expression.

1. (c'){c){c') 2. m

m •t\3. (V)

4. c/" 5. {p'q'){p\f) 6. 45yW

5/z

7. {-ry 8. 3 , /V" 9. {4h'k')(l5k'h')

10. 11. Om'nY 12. ( 1 2 x V ) "

13. i-5u-b)(2ah-c)i-3b) 14. 4x(2x-; ;)" 15. {3x'y)(2y-y

4

D. Binomial Multiplication

I . Reviewing the Distributive Property

The distributive property is used when you want to muhiply a single term by an expression.

Ex I : 8(5x^ -9x)

8-5x^ + 8 - ( - 9 x )

40.V-- - 7 2 x

I I . Multiplying Binomials - the F O I L method

When multiplying two binomials (an expression with two terms), we use the " F O I L " method. The " F O I L " method uses the distributive property twice!

F O I L is the order in which you will multiply your terms.

First

Outer

Inner

Last

Ex. 1: (x + 6) (x+ 10)

First X X >

Outer > lOx

Inner 6-x > 6x

Last 6-10 > 6 0

X- + 10X + 6A- + 60

O U T E R

I N N E R L A S T

X - + 16x+60 (After combin ing lii<e terms)

Recall; 4 ' = 4 • 4

X' = X - X

Ex.

(x + 5)^ = (x + 5)(x+5) Now you can use the " F O I L " method to get a simpHfied expression.


Mul t ip ly . Write your answer in simplest form.

1. ( x + 1 0 ) ( x - 9 ) 2. (x + 7 ) ( x - 12)

3. ( x - 1 0 ) ( x - 2 ) 4. ( x - 8 ) ( x + 81)

5. ( 2 x - l ) ( 4 x + 3) 6. ( -2x+10)(-9x + 5)

7. ( - 3 x - 4 ) ( 2 x + 4) 8. (x + 10)'

9. (-x + 5)^ 10. ( 2 x - 3 ) -

E. Factoring

1. Using the Greatest Common Factor ( G C F ) to Factor.

• Always determine whether there is a greatest common factor (GCF) first.

E x . 1 -33x-'+90.v^

• In this example the GCF i s3x ' .

• So when we factor, we have 3 . x ^ ( x ' - 1 Ix + 30).

• Now we need to look at the polynomial remaining in the parentheses. Can this trinomial be factored into two binomials? In order to determine this make a list o f all o f the factors o f 30.

30 30

1 30 -1 -30 2 15 -2 -15 3 10 -3 -10 5 6 -5 -6

Since -5 + -6 = -11 and (-5)(-6) = 30 we should choose -5 and -6 in order to factor the expression.

• The expression factors into 3x- (x - 5)(x - 6)

Note: Not all expressions w i l l have a GCF. I f a trinomial expression does not have a GCF, proceed by trying to factor the trinomial into two binomials.

I I . Applying the difference of squares: a~ - Ir = {a -h\a + h)

Ex. 2 4x' - lOOx Since x^ and 25 are perfect squares separated by a 4x( V " - 25] subtraction sign, you can apply the difference o f two

^ ' squares formula. 4 x ( x - 5 ) ( x + 5)


Factor each expression.

1. 3x' + 6x

3. x ' - 2 5

5. g ' - 9 ^ + 20

7. Z - - 7 2 - 3 0

9. 4y'-36y

2. 4a'h- -\6ab' +Sab'

4. + 8 n + 15

6. J ' + 3 f / - 2 8

8. w' + i8m + 8 l

10. 5k-+30k-\35

10

F. Radicals

To simplify a radical, we need to find the greatest perfect square factor o f the number under the radical sign (the radicand) and then take the square root o f that number.

Ex. 1: a/72

V 3 6 - V 2

6^2

Ex. 2 : 4^90

4 - V 9 - V l O

4-3-VlO

I2V1O

£x . 3: V48

A/I6V3

4A/3

OR This is not simplified completely because 12 is divisible by 4 (another perfect square)


Simplify each radical.

1. V m 2. V 9 0 3. V 1 7 5 4. V 2 8 8

5. V 4 8 6 6. 2 V I 6 7. 6V500

8. 3V147 9. 8 V 4 7 5 10. 1 2 5

G. Graphing Lines

1. Finding tlie Slope of the Line that Contains each Pair of Points.

Given two points wi t l i coordinates {x],y\ and {xi^}'! ) ' ^^e formula for the slope, m, o f

y i - y\ the line containing the points is m = Xi - X |

Ex. (2, 5) and (4, 1) Ex. (-3, 2) and (2, 3) 1-5 - 4 , 3 - 2 1

m = = = - 2 w = 4 - 2 2 2 - ( - 3 ) 5

The slope is -2. The slope is ^


1. (-1,4) and (1 , -2) 2. (3, 5) and (-3, 1) 3. (1 , -3) and (-1,-2)

4. (2 , -4) and (6 , -4) 5. (2, 1) and (-2,-3) 6. (5 , -2) and (5, 7)

11. Using the Slope - Intercept Form of the Equation of a Line. The slope-intercept form for the equation o f a line wi th slope in and j^-intercept 6 is v = mx +

Ex. y = l)x-\

Slope: 3 v'-intercept:

1 1

—

T 1 1

1 i \

— n 1 1

— J t i

T ! /i __ 1 ZL

1 1

—

1 I /

1 1

—

1 1

_ _ H 1 1

-

1 1

—

1 1

/ t / 1

] / 1 k 1

1

1 J

^ 1 1

L J 1 1 i 1 1 H —

\ 1

- - + - -1 1

1 — —

r '

1 ' 1

Place a point on the jy-axis at - 1 . Slope is 3 or 3/1, so travel up 3 on the>'-axis and over 1 to the right.


1. v = 2x + 5

Slope: ^-intercept:

Ex. y = - — x + 2 4

Slope: - — v-intercept: 2

Place a point on the v-axis at 2. Slope is -3/4 so travel down 3 on the j^-axis and over 4 to the right. Or travel up 3 on the >'-axis and over 4 to the left.

1. y = - x - i

Slope: V-intercept:

3. V = - — X + 4 5

Slope: _

y-intercept: __

-I r - - i ^- - | r - - i ^-

-| ^-

_ i L _ _ l \

^ - - 1

I 1 -

I \

4. j = - 3 x

Slope:

y'-intercept

I— I

I I - I L I ^-

1 1

4 I — I

I — I I I

5. V = -x + 2 6. v = x

Slope:

>'-intercept:

Slope:

v-intercept

- T - - )

I 1 -

I l _

I

I

I ^-

I 1 J r L_

-| 1 -

- J -

y

I — I I — I u _

I i

1 1

I 1

I 1

I I

" T — I ^-

_ L _

I I I . Using Standard Form to Graph a Line.

A n equation in standard form can be graphed using several different methods. Two methods are explained below.

a. Re-write the equation m y = mx + h form, identify the v-intercept and slope, then graph as

in Part I I above.

b. Solve for the x- and y- intercepts. To find the x-intercept, let v = 0 and solve for x. To find the jv-intercept, let x = 0 and solve fory. Then plot these points on the appropriate axes and connect them wi th a line.

Ex. 2x-3y = \0 a. Solve for 7. OR b. Find the intercepts:

-3y = -2x + 10 let >- = 0 : let x = 0:

y= + 2 x - 3 ( 0 ) = 10 2 ( 0 ) - 3 y = 1 0

y = - x - — 2x = 10 - 3 j = 10

'0 X = 5 ~ ^

3 So x-intercept is (5, 0) So v-intercept is

15


1. 3 x + v = 3

y

[ r T r i . 1 r n T 1 - 1 • 1 1 1 n T 1

- f — | - " f — r 1 1

1 — 1 - T r T r 1 — ] _ I t - n T 1

] j _ [ 1 j 1 ] ] 1 1 I k . 1 — \ — 1 — r > -

] ]

_[ [_ \ i _ - f - | - —

1 — ^ - - 1 — r " - T - - r — I - - — r ~ r -I — T — 1 —

L J 1 1 . J L I

—[-1 J 1 '

' ]

2. 5x + 2y = IO

V 1 1 1 1

v

L J 1

1

1 -I—

1 ] ] 1 4 -\-1

1

1 - j - --}- " 1 " T " 1 -i- - - ] ' -

1

i 1

i 1

1

I -|__

1 ^ 1 j 1 I ^

1 " 1 1 I

1 1 1

1

! -\-

] - 1 - ] ] 1

1 1

1

] " f

1

1

1 r 1

1 - 1 - 1

1 1 1

1

1 1

3. y=^A 4. 4 x - 3 v = 9

1 — )

I I

I — I

I — I

I — I 1 — 1

•4 1 I

1 — I

1 — I

I I

I r

7

I I

I 1

1 I

1 1

1 1

11^

5. - 2 x + 6;' = 12

V i 1 1 1

r n - T 1 r 1 1 1

1 1 1

] 4

] 1 1 1

1 1 1

1 — 1 - - l -1

- I - _ _ [ _ - 1 - - - - 1 - - t -1 1 [

- H 1 - - -t 1 1 1

1

1 -1-

1 1 1 1

-1-1 1 1

1 1 1 1 1 1 1 1^ 1

1

-|-1 1 1

1 -]- " 1 " — j ] ]

1 1 1

1 — 1 -1

1 - t - - h ~ H - - - 1 - - f - 1 - - H -

1 1 1 - -(- - - 1 - - ̂

1

1 1

- r --\~ - ] -

L J 1 1 1 1 1 1 1 1 i J

1 1 1 1 1 1 1 1 1

6. A - = -3

y 1 ] ] [

i 1 ]

r- 1 1 T -l—i

— -\—\ 1

1

1

1 t (

] ] [ - 1 ] 1

( 1

1 4 ^

1 1

— 1

1 1 1 1 1 1 1 1 ^ 1 1 ^ 1 1 1 j

i W\ 1

1 ] -\

] 1 1

1

1 [

- f -|- [ [ r

1 1 -1

- r -1

- r • - T - - r -1

- r - 1 1

r n 1 1 1 -{- 1 1 - i - I 1 n

1 r 1 1

1 1 1 1 - j -

1

1 i— 1

j - 1 _ i _ - - l _ 1 1 _ i I

Algebra 2 Summer Review Packet Student Answer Key

A. Simplifying Polynomial Expressions


1. 24x + 3y 2. - 15.v^ + 37.v + 22

3. 9 « - 3 4. -22A + 6

5. 160vjc+n0( / 6. -5JC + 6

7. 74z-24w 8. 56c--117

9. -27JC^+54JC-9 10. - J + 31A- + 42

B. Solving Equations


1. A--7 2. x - 2 6

3. jc = 9.5 4. A: = 13

5. A: = 3.5 6. x = 1 6

7. x = 19 8. jc = -2.8

9. x = 9.5 10. jc = 0


1. V^W-Y 2. H ' = -

3. / = = - 3 + - r f 4. A- = = -3 3 d d d

P+\(,1Q P ^ , 9y + u + 5lt 5. ? = = + 9 6. x = —

180 180 4

C . Rules of Exponents


I. c

4. 1

7. 21

10. 3c

13. mi^h-^c

2. m 12

5. p''<r

• 3 8. 3 /

11. Slm^n"^

14. 4x

3. k''

6. 9z'̂

9. 60/;**^

12. 1

15. 24A--*

D. Binomial Multiplication


1. + A : -90

3. - 12X + 2 0

5. Hx^ +2x-3

1. -6.\ - 2 0 x : - 1 6

9. 25-IO.v + x^

2. X - 5 X - 8 4

4. x^ + 73X -648

6. 50-100x+l8x^

8. x^ + 20x+ 100

10. 4x^ - 12x + 9

E . Factoring


1. 3x (x+2)

3. ( x - 5 ) ( x + 5)

5. (A' -4 )a ' -5 )

7. ( z -10 ) ( z + 3)

9. 4 j (v -3 ) ( .v + 3)

2. 4uh^(a - 4h + Ic)

4. (/» + 5)(/; + 3)

6. ((/ + 7 ) ( r / -4 )

8. (m + 9)^

10. 5 (A+9)(A-3)

F . Radicals


1. 11 2. 3^10 3. 5^7 4. 12V2

5. 9^6 6. 8 7. 60V5 8. 21V3

9. 40Vl9 10. ^

G . Graphing Lines


1. -3 2. I 3. - i 3 2

4. 0 5. 1 6. undefined

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