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Download Algebra 3 Summer Assignment 2016
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Algebra III Summer Assignment 2016 The following packet contains topics and definitions that you will be required to know in order to succeed in Algebra III this year. You are advised to be familiar with each of the concepts and to complete the included problems by Thursday, September 1, 2016. All of these topics were discussed in either Algebra II or Geometry and will be used frequently throughout the year. All problems that you are to complete are marked in bold. All problems are expected to be completed. Section I: Lines Lines: 𝑦 −𝑦 Slope: 𝑚 = 𝑥2 −𝑥1 Slope Intercept Form: y = mx + b Standard Form: ax + by = c Point-Slope Form: y - y1 = m(x – x1) 2 1 Intercepts: In order to find the x-intercept(s) of an equation, you have to set y equal to zero and solve the equation for x. In order to find the y-intercepts of an equation, you have to set x equal to zero and solve the equation for y. Parallel Lines: Two lines whose graphs have the same slope. Perpendicular Lines: Two lines whose graphs have opposite reciprocal slopes. Find the slope of the lines passing through each set of points: 𝟏 (𝟔, 𝟏𝟐) 𝒂𝒏𝒅 (−𝟔, −𝟐) Find the slope of the line: 𝟏 𝟏 (− 𝟑 , 𝟎) 𝒂𝒏𝒅 (− 𝟐 , − 𝟐) Find the F Find the slope of the following Line: y 𝟑𝒙 + 𝟓𝒚 = −𝟏𝟓 4 2 –4 –2 O –2 –4 2 4 x Write the equation of the line in standard Find the point-slope form of the line form for a slope of -8 and through (-2, -2) through (-6, -4) and (2, -5). Write the equation of the line in slope-intercept Write the equation of the horizontal form through (−𝟏𝟎, −𝟑) and (−𝟏, 𝟏). line through (−𝟑, −𝟏). Graph the equation of 6x + 5y = 30 using the x- and y-intercepts. 𝟑 𝟒 Graph 𝒚 = − 𝒙 − 𝟐 Find the equation of the line that is parallel Find the equation of the line perpendicular to 𝒚 = − 𝟑𝒙 + 𝟒 and goes through (−𝟒, 𝟔). to 𝒚 = − 𝟒 𝒙 + 𝟏 and goes through (2, 6). 𝟓 Give the slope-intercept form for the equation of the line that is perpendicular to 𝟕𝒙 + 𝟑𝒚 = 𝟏𝟖 and contains (𝟔, 𝟖). Which two lines are parallel? I. II. III. What must be true about the slopes of two perpendicular lines, neither of which is vertical? Section II: Polynomials Properties of Exponents: 𝑥 𝑛 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ … ∙ 𝑥 (n factors of x) 1. Whole number exponents: 𝑥 0 = 1, 𝑥 ≠ 0 2. Zero exponents: 𝑥 −𝑛 = 3. Negative Exponents: 1 𝑥𝑛 𝑛 √𝑥 = 𝑎 → 𝑥 = 𝑎 𝑛 4. Radicals (principal nth root): 5. Rational exponents: 𝑥 1⁄ 𝑛 6. Rational exponents: 𝑥 𝑚⁄ 𝑛 𝑛 = √𝑥 𝑛 = √𝑥 𝑚 Operations with Exponents: 𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑚+𝑛 1. Multiplying like bases: 𝑥𝑚 2. Dividing like bases: 𝑥𝑛 = 𝑥 𝑚−𝑛 𝑥 𝑛 𝑥𝑛 3. Removing parentheses: (𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛 ( ) = 𝑛 (𝑥 𝑛 )𝑚 = 𝑥 𝑛𝑚 𝑦 𝑦 For example: Simplify each of the following expressions: a. 7a 2a 2 5 7 2a ( 2 5) 14a 3 14 3 a b. 2 x 1 y2 y 2 x 1 3 3 8 x y 8y x3 6 c. 3 6 3 2ab 5 c 2 a 3bc 2 2 3 2a (13)b ( 51) c ( 2 2 ) 2a 2b 4 c 0 2b 4 2 a Simplify the following expressions. Use only positive exponents. 1. 3a 4a 2. 4x 2x 2 6 13. x 4 x 2 x 5 2 2 14. 12 x y 15. 4 p q p q 16. 4x3 2x 17. p 18. 15 x 4 3x 19. r 2 s 3t 4 r 2 s 4 t 4 20. xy 2 6 x 2 y2 21. s t st 2 3. 4 x y 4. 2 x 5 2 3 5 y 4 3 5 5. 6. 8a 2a 2 6x 7 y 5 3x 1 4 x 2 0 7. 8. 9. 2 xy5 3x 2 2 2 6m n 3mn 2 6 2 8x 4 y 7 2 2 2 2 2 10. 3x 4 y 5 2 3 3 22. 3x 3 y 2 2 11. 2r 1 2 0 2 s t 2rs 23. 12. x 5 2x 3 24. h k 4 5 0 s 2t 3 sr 3 r t 3 Section III: Radical Expressions and Complex Numbers Radical Expressions: 𝑛 𝑛 𝑛 𝑛 𝑛 Multiplying Radical Expressions: If √𝑎 𝑎𝑛𝑑 √𝑏 are real numbers, then √𝑎 ∙ √𝑏 = √𝑎𝑏 𝑛 𝑛 𝑛 Dividing Radical Expressions: If √𝑎 𝑎𝑛𝑑 √𝑏 are real numbers and b≠ 0, then √𝑎 𝑛 𝑎 = √𝑏 √𝑏 𝑛 For example: Simplify each of the following rational expressions (be sure to rationalize the denominator if necessary): √𝑥 3 √5𝑥𝑦 First use the Dividing Radical Expressions rule to rewrite with one radical. √ Next cancel any common factors in the numerator and denominator. 𝑥3 5𝑥𝑦 𝑥2 √ 5𝑦 √𝑥 2 Now use the Dividing Radical Expressions rule to rewrite with two radicals. √5𝑦 𝑥 Simplify any possible square roots. √5𝑦 𝑥 √5𝑦 ∙ √5𝑦 √5𝑦 Rationalize the denominator to get you final answer. 𝑥√5𝑦 5𝑦 Final Answer Simplify the following expressions. Use only positive exponents. 1) √2 5) √3 2 2) √3𝑥 6) √2𝑥 3 7) 3) 4) √10𝑥𝑦 15√60𝑥 5 3√12𝑥 8) √3𝑥𝑦 2 √5𝑥𝑦 3 √5𝑥 4 𝑦 √2𝑥 2 𝑦 3 10 √5𝑥 2 3√11𝑥 3 𝑦 −2√12𝑥 4 𝑦 Imaginary Numbers: For any positive real number a, √𝑎 = 𝑖 √𝑎 𝑖 2 = −1 𝑜𝑟 √−1 = 𝑖 For example: Simplify √−8 by using the imaginary number i. √−8 First factor out the -1 from -8 √−1 ∙ 8 𝑖√8 Next replace √−1 with i 𝑖√4 ∙ 2 Then factor any perfect squares. 2𝑖√2 Take the square root of the perfect square. Simplify the following by using the imaginary number i. 1) √−12 4) √−68 2) √−20 5) √−1 3) √−300 6) √−121 Complex Numbers: A complex number can be written in the form 𝑎 + 𝑏𝑖, where a & b are real numbers, including 0. For example: Simplify the expression (5 + 7𝑖) + (−2 + 6𝑖) (5 + 7𝑖) + (−2 + 6𝑖) Use the Commutative and Associative Properties of Addition to rewrite expression with like terms together (5 + −2) + (7𝑖 + 6𝑖) (3 + 13𝑖) Simplify For example: Simplify the expression (5𝑖 + 1)(−4𝑖) (5𝑖 + 1)(−4𝑖) (5𝑖)(−4𝑖) + 1(−4𝑖) Use the Distributive Property to rewrite expression. −20𝑖 2 − 4𝑖 Multiply Replace 𝑖 2 with -1 −20(−1) − 4𝑖 20 − 4𝑖 Simplify Simplify the following by using the imaginary number i. 7) (2 + 4𝑖) + (4 − 𝑖) 10) 6 − (8 + 3𝑖) 8) (12 + 5𝑖) − (2 − 𝑖) 11) (8 + 𝑖)(2 + 7𝑖) 9) (−2𝑖)(5𝑖) 12) (9 + 4𝑖)2 Section IV: Quadratic Equations Standard Form of a Quadratic Equation: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 Sum and Difference of Same Terms: (𝑢 + 𝑣)(𝑢 − 𝑣) = 𝑢2 − 𝑣 2 Square of a Binomial: (𝑢 + 𝑣)2 = 𝑢2 + 2𝑢𝑣 + 𝑣 2 and (𝑢 − 𝑣)2 = 𝑢2 − 2𝑢𝑣 + 𝑣 2 For example: Write the following quadratic equations in standard form, then identify a, b, and c. 2 First move the −3𝑥 to the other side of the equation by adding 3𝑥 2 to both sides of the equation Now move the 2 to the other side of the equation by subtracting 2 from both sides. So a=3 b=4 and c=-2 4𝑥 = 2 − 3𝑥 2 +3𝑥 2 + 3𝑥 2 3𝑥 2 + 4𝑥 = 2 -2 -2 3𝑥 2 + 4𝑥 − 2 = 0 Write the following quadratic equations in standard form, then identify a, b, and c. 1) 2𝑥 2 − 11𝑥 = −15 4) 2𝑥 2 − 1𝑥 = −5 − 3𝑥 2) 𝑥 2 + 7𝑥 = 18 5) −7𝑥 = 6𝑥 2 − 4 3) 16𝑥 2 = 8𝑥 6) −14𝑥 − 2𝑥 2 = 6𝑥 2 + 3𝑥 For example: Solve the following quadratic equations by factoring: 𝑥 2 + 5𝑥 + 4 = 0 To factor this quadratic equation, you first must split the 𝑥 2 term. (𝒙 ) (𝒙 ) =𝟎 Now you must find two numbers that multiply to equal c (4) Either 1 * 4 or 2 * 2 will work. You must choose the set of numbers that and add to equal b (5) In this case 1 & 4 Set each set of parenthesis = 0 to and solve (𝒙 + 𝟏 ) (𝒙 + 𝟒) =𝟎 (𝒙 + 𝟏 ) = 𝟎 𝒐𝒓 (𝒙 + 𝟒 ) = 𝟎 𝒙 = −𝟏 𝒐𝒓 𝒙 = −𝟒 Solve the following quadratic equations by factoring: Since the c=0 in the equation, you factor this quadratic by taking out the GCF −3𝑥 2 + 12𝑥 = 0 The GCF of both terms is -3x, so factor this out of BOTH terms. (Note: you can check you factored correctly by re-distributing -3x, you should get the original equation.) (−𝟑𝒙)(𝒙 − 𝟒) = 𝟎 Lastly, set each set of parenthesis = 0 to and solve −𝟑𝒙 = 𝟎 𝒐𝒓 𝒙 − 𝟒 = 𝟎 𝒙 = 𝟎 𝒐𝒓 𝒙 = 𝟒 Factor each completely: 7) 18 x3 12 x 2 6 x 4 8) 64k 3 56k 2 32k 28 9) 8k 3 40k 2 32k 160 Factor the common factor out of each expression: 10) 15 30 x 10 x 2 11) 20n 4 35n 2 35n Factor each completely: 12) v 2 3v 2 13) v 2 9v 18 14) x 2 5 x 14 15) 2n 2 7v 16) 7 x 2 50 x 7 17) 28n 2 240n 108 18) 8k 2 36k 19) 10b 2 12b 20) 10n 2 14n 21) 27a 2 36a 12 22) 75r 2 48 23) 9 x 2 12 x 4 24) n 2 1 25) 48a 2 27 26) 25n 2 9 Solve each of the following by using the Quadratic Formula: 𝑥 = 27) 𝒙𝟐 + 𝟐𝒙 + 𝟒 = 𝟎 29) −𝑏±√𝑏2 −4𝑎𝑐 2𝑎 28) 𝟑𝒙𝟐 + 𝟐𝒙 + 𝟏 = 𝟎 𝟖𝒙𝟐 − 𝟏𝟎𝒙 − 𝟑 = 0 30) 𝒙𝟐 − 𝟐𝒙 − 𝟑 = 𝟎 Expand each of the following (hint: FOIL): 31) (𝒙 + 𝟒)𝟐 32) (𝒙 − 𝟐)𝟑 33) (𝒙 − 𝟒)(𝒙 + 𝟑) 34) (𝒙 − 𝟔)(𝒙 − 𝟑)