Algebra 2 Summer Homework Packet 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 when you start the term so consider this your warm up lap before the sprint starts. 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 6^^ grader. Show me every step. 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: Cheers, Mrs. Kowalski www.nmthforum.orq/library/drmath/drmath.middle.html www.purplemath.com www.virtualnercl.com www.math.com www.freemathhelp.com/alrebra-help
A. Simplifying Polynomial Expressions. Combining Like Terms - You can add or subtract terms that are considered "Hlce", or terms that have the same variable(s) with the same exponent(s). Ex. : 5x-7y+ 0x + 3y 5x;;Jy + lox + ly 5x-4y Ex.2: -8h^+lOh'- 2h^-5h^ -8h^+ loh^- 2ir^- 5h^ -20h^ - 5h^ L Applying the Distributive Property Every tenn inside the parentheses is multiphed by the term outside of the parentheses. ^-x. : 3(9x-4) Ex.2: Ax^{5x^+6x) 3 9A: - 3 4 4x' 5x' + 4x- 27X-2 20x'+24x-' L Combining Like Terms AND the Distributive Property (Problems with a Mix!) - Sometimes problems will require you to distribute AND combine like terms!! x. l:3(4x-2) + 3x 3-4x-3-2+3x x. 2 : 3(2x - 5) - 9(-7 + lox) 3-2x-3-5-9(-7)-9(0x) 2x-6 + 3x 36x-5 + 63-90x 25x-6-54X + 48
PRACTCE SET Simplify. 2. 4v + 22-5v^ +23.V. 8x-9y + \6x + \2y 4. -2(/)-3) 3. 5n-{3-4n) 6. -(5x-6) 5. ]Qq(\6x + \\) 8. (8c + 3)+i2(4c'-i0) 7. 3(8z-4w)+2(0z-6w) 9. 9(6x-2)-3(9x^ -3) 0. -(v-x) + 6(5x + 7) 2.
. Solving Two-Step Equations B. Solving Equations A couple of hints:. To solve an equation, UNDO the order of operations and work in the reverse order. 2. REMEMBER! Addition is "undone" by subtraction, and vice versa. Multiplication is "undone" by division, and vice versa. x. : 4x-2 = 30 x. 2 : 87 =- LY + 2 + 2 +2-2 -2 4x = 32 66 = -llx ^4 +4 ^- ^- X = 8-6 = X. Solving Multi-step Equations With Variables on Both Sides of the Equal Sign When solving equations with variables on both sides of the equal sign, be sure to get all terms with variables on one side and all the terms without variables on the other side. Ex. 3; 8x + 4 = 4x + 28-4 -4 8x =4x + 24-4x -4x 4x = 24 ^4 +4 x = 6. Solving Equations that need to be simplified first n some equations, you will need to combine like terms and/or use the distributive property to simplify each side of the equation, and then begin to solve h. ^x.4; 5(4x-7) = 8x + 45 + 2x 20x-35 = 0x + 45 -lox -lox 0x-35 = 45 + 35 +35 0x = 80 +0 +0 x = 8 3
PRACTCE SET 2 Solve each equation. You must show all work.. 5x- 2 = 33 2. 40 = 4x + 36 3. 8(3x-4) = 96 4. 45.x-- 720 + 5x = 60 5. l32 = 4(2x-9) 6. 98 = 54 + 7A--68 7. -3 = -5(3A--8) + 6X i. - 7A--0 = 8+ 3.Y 9. 2x + 8-l5 = -2(3x-82) 0. -(2x-6) = i2x + 6 V. Solving Literal Equations - A literal equation is an equation that contains more than one variable. - You can solve a literal equation for one of the variables by getting that variable by itself (isolating the specified variable). Ex. : 3xv = 8, Solve for x. 3xy 8 3 V " 3y _6_ V Ex. 2: 5a-\0b = 20, Solve for a. + 0/) =+ 0/? 5a = 20+06 5a 20 0/7 = + 5 5 5 a = A + 2b
PRACTCE SET 3 Solve each equation for the specified variable.. Y+V=W, for K 2. 9wr=8,for w 3. 2^-3/= 9, for/ 4. clx + t= 0, for X 5. P = (g-9)80, forg 6. 4.r + ;^ - 5/2 = Oy + u, for x
C. Rules of Exponents Multiplication: Recall (x'")ix") = x^"'^"' 'x:(3x V)(4x/)=(3 4)(x' x')(v" /) = 2A--V" Division: Recall = x""-"' x" Ex: 42m\r - 3m\/ 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 of Zero: Recall x" = \, x ^ 0 Ex: 5x%' = (5)()(/) = 5 PRACTCE SET 4 Simplify each expression.. (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') 0.. Om'nY 2. (2xV)" 3. i-5u-b)(2ah-c)i-3b) 4. 4x(2x-;;)" 5. {3x'y)(2y-y 4
D. Binomial Multiplication. Reviewing the Distributive Property The distributive property is used when you want to muhiply a single term by an expression. Ex : 8(5x^ -9x) 8-5x^ +8-(-9x) 40.V-- -72x. Multiplying Binomials - the FOL method When multiplying two binomials (an expression with two terms), we use the "FOL" method. The "FOL" method uses the distributive property twice! FOL is the order in which you will multiply your terms. First Outer nner Last Ex. : (x + 6)(x+ 0) OUTER First X X > Outer > lox nner 6-x > 6x NNER LAST Last 6-0 >60 X- + 0X + 6A- + 60 X - + 6x+60 (After combining lii<e terms)
Recall; 4'= 4 4 X' = X - X Ex. (x + 5)^ = (x + 5)(x+5) Now you can use the "FOL" method to get a simphfied expression. PRACTCE SET 5 Multiply. Write your answer in simplest form.. (x+ 0)(x-9) 2. (x + 7)(x- 2) 3. (x- 0)(x-2) 4. (x-8)(x + 8) 5. (2x-l)(4x + 3) 6. (-2x+0)(-9x + 5) 7. (-3x-4)(2x + 4) 8. (x + 0)' 9. (-x + 5)^ 0. (2x-3)-
E. Factoring. Using the Greatest Common Factor (GCF) to Factor. Always determine whether there is a greatest common factor (GCF) first. Ex. -33x-'+90.v^ n this example the GCF is3x'. So when we factor, we have 3.x^(x'- x + 30). Now we need to look at the polynomial remaining in the parentheses. Can this trinomial be factored into two binomials? n order to determine this make a list of all of the factors of 30. 30 30 30 - -30 2 5-2 -5 3 0-3 -0 5 6-5 -6 Since -5 + -6 = - 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 will have a GCF. f a trinomial expression does not have a GCF, proceed by trying to factor the trinomial into two binomials.. Applying the difference of squares: a~ - r = {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 of two ^ ' squares formula. 4x(x-5)(x + 5)
PRACTCE SET 6 Factor each expression.. 3x' + 6x 2. 4a'h- -\6ab' +Sab' 3. x'-25 4. +8n + 5 5. g'-9^ + 20 6. J' + 3f/-28 7. Z--72-30 8. w' + i8m + 8l 9. 4y'-36y 0. 5k-+30k-\35 0
F. Radicals To simplify a radical, we need to find the greatest perfect square factor of the number under the radical sign (the radicand) and then take the square root of that number. Ex. : a/72 V36-V2 6^2 Ex. 2: 4^90 4-V9-VlO 4-3-VlO 2VO x. 3: V48 A/6V3 4A/3 OR This is not simplified completely because 2 is divisible by 4 (another perfect square) PRACTCE SET 7 Simplify each radical.. V m 2. V90 3. V75 4. V288 5. V486 6. 2V6 7. 6V500 8. 3V47 9. 8V475 0. 25
G. Graphing Lines. Finding tlie Slope of the Line that Contains each Pair of Points. Given two points witli coordinates {x],y\ and {xi^}'! )' ^^e formula for the slope, m, of yi - y\ the line containing the points is m = Xi - X Ex. (2, 5) and (4, ) Ex. (-3, 2) and (2, 3) -5-4, 3-2 m = 4-2 = 2 = -2 w = 2-(-3) 5 The slope is -2. The slope is ^ PRACTCE SET 8. (-,4) and (,-2) 2. (3, 5) and (-3, ) 3. (,-3) and (-,-2) 4. (2,-4) and (6,-4) 5. (2, ) and (-2,-3) 6. (5,-2) and (5, 7)
. Using the Slope - ntercept Form of the Equation of a Line. The slope-intercept form for the equation of a line with slope in and j^-intercept 6 is v = mx + Ex. y = l)x-\ Slope: 3 v'-intercept: Ex. y = - x + 2 4 Slope: - v-intercept: 2 T J t i H ] J i H r i \ T! /i _ ZL / t / / / k ^ \ - - + - - ' ' - n L J Place a point on the jy-axis at -. Slope is 3 or 3/, so travel up 3 on the>'-axis and over to the right. 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. PRACTCE SET 9. v = 2x + 5. y = - x - i Slope: ^-intercept: Slope: V-intercept:
3. V = - X + 4 5 Slope: y-intercept: 4. j = -3x Slope: y'-intercept - r--i ^- - r--i ^- - ^- - L ^- _i L l \ 4 ^ - - - \ 5. V = -x + 2 6. v = x Slope: >'-intercept: Slope: v-intercept - l_ - T - - ) u _ y " T ^- _ L _ i ^- - - - J - J r L_
. Using Standard Form to Graph a Line. An 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 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 with a line. Ex. 2x-3y = \0 a. Solve for 7. OR b. Find the intercepts: -3y = -2x + 0 let >- = 0 : let x = 0: y= + 2x-3(0) = 0 2(0)-3y=0 y = -x- 2x = 0-3j = 0 '0 X = 5 ~ ^ So x-intercept is (5, 0) So v-intercept is 3 5
PRACTCE SET 0. 3x+v = 3 2. 5x + 2y = O y [ r T r i. r n T - n T -f - - T r T r " f r ] _ ] j _ [ j t- n T _[ [_ \ i_ ] ] - f - - ^ - - r " - T--r - - r ~ r - T L J. J L ] ] k. \ r>- [- ' ] J ' L J V - j - --}- "" T " -i- --]'- i i v - - ^ j ^ " ] ] 4 -\-! ] ] ] -\- -- ] " f r -- 3. y=^a 4. 4x-3v = 9 r ) 7 4 ^
5. -2x + 6;' = 2 r n - T r - - l - - V i ] ] 4 - - [ _ - - - - - - - t - [ - H - - -t -- -]- "" -- - - j ] ] ^ - t - - h ~ H - - - - - f - - - H - - -(- - - - - ^ L J i J 6. A-= -3 y ] ] [ i ] t ( ] ] [ r- - T ] ( 4^ -l i -\ \ ^ ^ j i W\ - r --\~ -]- ] ] -\ [ - f - - [ [ r --r- -r - T - -r - r n -{- -i- n i -r- r -j- j- _ i _ - - l i
Algebra 2 Summer Review Packet Student Answer Key A. Simplifying Polynomial Expressions PRACTCE SET. 24x + 3y 2. - 5.v^ + 37.v + 22 3. 9«-3 4. -22A + 6 5. 60vjc+n0(/ 6. -5JC + 6 7. 74z-24w 8. 56c--7 9. -27JC^+54JC-9 0. -J + 3A- + 42 B. Solving Equations PRACTCE SET 2. A--7 2. x-26 3. jc = 9.5 4. A: = 3 5. A: = 3.5 6. x=6 7. x = 9 8. jc = -2.8 9. x = 9.5 0. jc = 0 PRACTCE SET 3. V^W-Y 2. H ' = - 3. /= = -3+-rf 4. A- = = -3 3 d d d P+\(,Q P ^, 9y + u + 5lt 5.? = = + 9 6. x = 80 80 4
C. Rules of Exponents PRACTCE SET 4. c 2. m 2 3. k'' 4. 5. p''<r 6. 9z'^ 7. 2 3 8. 3/ 9. 60/;**^ 0. 3c. Slm^n"^ 2. 3. mi^h-^c 4. 4x 5. 24A--* D. Binomial Multiplication PRACTCE SET 5. + A:-90 2. X -5X-84 3. - 2X + 20 5. Hx^ +2x-3 4. x^ + 73X-648 6. 50-00x+l8x^. -6.\ -20x:-6 8. x^ + 20x+ 00 9. 25-O.v + x^ 0. 4x^ - 2x + 9 E. Factoring PRACTCE SET 6. 3x(x+2) 3. (x-5)(x + 5) 5. (A'-4)a'-5) 7. (z-0)(z + 3) 9. 4j(v-3)(.v + 3) 2. 4uh^(a - 4h + c) 4. (/» + 5)(/; + 3) 6. ((/ + 7)(r/-4) 8. (m + 9)^ 0. 5(A+9)(A-3)
F. Radicals PRACTCE SET 7. 2. 3^0 3. 5^7 4. 2V2 5. 9^6 6. 8 7. 60V5 8. 2V3 9. 40Vl9 0. ^ G. Graphing Lines PRACTCE SET 8. -3 2. 3. -i 3 2 4. 0 5. 6. undefined
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