Honors Geometry/Trigonometry course 4524/4525 Desert Mountain High School Math Department Summer Work Packet Honors/AP/IB level math courses at Desert Mountain are for students who are enthusiastic learners of mathematics and whose work ethic is of the highest standard. These students are expected to arrive ready to go on the first day of school. The attached packet is designed to help you review concepts with which you should already be familiar. It is recommended that you complete some of the problems from the packet at the beginning of the summer when the concepts are still fresh, and then complete the remainder of the problems near the beginning of the school year. If you do not complete the problems in the packet, your grade will not be affected directly, however, the material in the packet has been taught in your previous math classes and will be assumed to be fully understood by you. I, the teacher, strongly advise you to work the problems this summer. The problems will be collected and reviewed after the first week of school once we have gone over any questions you have. If you are new to the Scottsdale Unified School District and did not receive notice of this assignment until registration, these review problems will be checked at the end of August. Some suggestions for the presentation and completion of mathematics assignments at DMHS are listed below. If you adhere the these guidelines with your summer work, you will be ready to meet the expectations of your mathematics teacher during the school year. Use notebook paper or plain white paper if extra paper is needed All work should be neat and legible Neatly place completed work in a small binder or folder with brads Use pencil, erase completely when needed Work the problems in order and clearly indicate section and problem numbers if working on a separate sheet of paper Begin new sections on a new piece of paper
ALGEBRA 1 REVIEW SECTION
Factoring Practice I. Factoring is the reverse of multiplying. To factor a polynomial, factor out the greatest common factor of each term. Example: 6a 2 b 12ab 2 = 6ab(a 2b) Factor completely. 1.x 2-3x 2. 6y 3-12y 2 + 3y 3. 8x 6 32x 5 + 16x 4 II.A binomial is a difference of two squares if both terms of the binomial are squares and there is a minus sign between the two terms. The difference of two squares, A 2 B 2, factors as two binomials (A B)(A + B). Which of the following are differences of squares? 4x 2 8y 2 no! -25 + 81a 2 yes (think commutative property) Factor completely. (This might take more than one step, look for common factors first!) 1. 9x 2 4 2. 4x 2 25 3. 2x 2 50 4. x 4 81 5. 16x 4 1 III. A trinomial square has three terms and is the square of a binomial. A 2 + 2AB + B 2 = (A+B) 2 and A 2-2AB + B 2 = (A-B) 2 Which of the following are trinomial squares? y 2 + 3y + 9 no! c 2-12c + 36 yes! 49a 2 56a + 16 yes, but why? 4x 2 4x 1 no, this is a tricky one Factor completely. 1. x 2 + 6x + 9 2. x 2-14x + 49 3. 9x 2-30x + 25 4. 25y 2-20y + 4 5. 18x 2 + 12x + 2 IV. To factor a trinomial of the type x 2 + bx + c, think of FOIL in reverse. Which of the following is the correct factorization of x 2 - x - 12? a) (x + 3)(x + 4) b) (x - 12)(x + 1) c) (x - 4)(x + 3) Only c) is correct. Factor completely. 1. x 2-8x + 15 2. x 2 + 4x -12 3. y 2 + 9y + 20 4. b 2-3b 18 5. p 2 7p 8 V. To factor a trinomial of the type ax 2 + bx + c, first check for common factors. Then test factors of the first and last terms to find the correct combination, using FOIL to test possible factorizations. Factor completely. 1. 2x 2-7x - 4 2. 6y 2-5y + 1 3. 4m 2 + 19m - 30 4. 6a 2-28a - 48
VI. A polynomial with four terms can sometimes be factored by grouping and using the distributive property twice. For example, a 3 + 2a 2 + 3a + 6 = (a 3 + 2a 2 ) + (3a + 6) = a 2 (a + 2) + 3(a + 2) = (a + 2)(a 2 + 3) Factor completely. 1. x 3 + x 2 + 3x + 3 2. x 4 + 4x 3-2x 8 3. 6x 3 + 4x 2 + 3x + 2 4. 9a 3-15a 2-12a + 20 1 Factoring: A General Strategy 1. Always look first for a common factor. 2. Then look at the number of terms. Two terms: Determine whether you have a difference of two squares Three terms: Determine whether the trinomial is the square of a binomial (a trinomial square, section III.) Four terms: Try factoring by grouping 3. Always factor completely. This might take several steps. EXAMPLES 10x 3 40x Look for a common factor. t 4 16 Factor the difference of two squares. 10x(x 2 4) Factor the difference of two squares. (t 2 + 4)(t 2 4) Factor the difference of two squares, again. 10x(x 2)(x + 2) Is is factored completely? Yes. (t 2 + 4)(t + 2)(t 2) x 4 10x 2 +25 This is a trinomial square! 3y 2 3y 6 Look for a common factor. (x 2 5) 2 It factors into a squared binomial. 3(y 2 - y - 2) No special model, think FOIL. 3(y - 2)(y + 1) VII. Factor the following problems completely. Show all the steps. 1. 2x 2-128 2. a 2 + 25 loa 3. 2x 2 llx + 12 4. x 3 + 3x 2-4x - 12 5. 24x 2-54 6. 20x 3 4x 2 72x 2
VII. continued Factor the following problems completely. Show all the steps. 7. 3t 2 27 8. y 2 + 49 + l4y 9. 8c 2-18c 5 10. x 3-5x 2-25x + 125 11. 9x 3 + 12x 2-45x 12. 8y 2 98 13. t 2 + 25 14. x 4 + 7x 3-3x 2-21x 15. 5x 5-80x 16. x 2 + 3x + 1 17. l - y 8 18. x 6-2x 5 + 7x 4 19. a 4-2a 2 + 1 20. 45-3x - 6x 2 21. 18 + y 3-9y - 2y 2 22. m 2 n 2 + 7mn 3 + 10n 4 23. 20-6x - 2x 2 24. a 2-1 9 25. x 3-18x 2 + 81x 26. a 2-5ab -14b 2 27. 3x 3 y - 2x 2 y 2 + 3x 4 y - 2x 3 y 2 28. 625x 4 16 29. x 2 + x + 1 4 30. x 3 + 24x 2 + 144x 3
Solving Systems of Equations There are several methods for solving a system of equations; graphing, substitution, and a third way that comes with different names. For the sake of this packet, it will be called the addition method. This method is especially helpful when both equations are written in standard form, Ax + By = C. EXAMPLE 1 Solve using the addition method. x + y = 5 x y = 1 Add the two equations together to eliminate one variable. x + y = 5 + x - y = 1 2x = 6 x = 3 Substitute 3 for x in either of the original equations to find y. x + y = 5 3 + y = 5 3-3 + y = 5-3 The solution of the system is the point of intersection of the two lines, (3,2). EXAMPLE 2 Solve using the addition method. 3x+2y=8 3x + y =7 Subtract the equations to eliminate one variable/ 3x + 2y = 8 3x + 2y = 8 -(3x + y = 7) -3x y = -7 y = 1 Substitute 1 for y in either of the original equations to find x. 3x + 2y = 8 3x + 2(l) = 8 3x + 2 = 8 3x +2 2 = 8-2 3x = 6 x = 2 The solution of the system is the point of intersection, (2,1). EXAMPLE 3 Solve using the addition method. 3x + 6y = -6 5x 2y = 14 Multiply the second equation by 3 so that y will be eliminated when the two equations are added together. 3x + 6y = -6 3x + 6y = -6 3(5x - 2y) = 3(14) + 15x - 6y = 42 18x = 36 x = 2 Substitute 2 for x in either of the original equations to find y. 3x + 6y = - 6 3(2) + 6y = - 6 6 + 6y = -6 6 6 + 6y = -6-6 6y = - 12 y = -2 The solution of the system is the point (2, 2). CHECK YOUR SOLUTION The solution must satisfy both equations. Check the solution for EXAMPLE 3 above. Replace x with 2, and y with -2 in both equations. 3x + 6y = -6 3(2) + 6(-2) = -6 6 + (-12) = -6-6 = -6 5x - 2y = 14 5(2) - 2(-2) = 14 10+4= 14 14 = 14 4
Solve the systems of equations using the addition method. Show all steps. 1. x y=7 x+y=3 2. x + y = 8 -x + 2y = 7 3. 3x - y = 9 2x + y = 6 4. 7c + 5d = 18 c - 5d = -2 5. 8x - 5y = -9 3x +5y = -2 6. x + y = -7. 3x + y = -9 7. x + y = 5 5x - 3y = 17 8. x - y = 7 4x - 5y = 25 9. 7p + 5q = 2. 8p - 9q = l7 10. 2a + 3b = -1 3a + 5b= -2 11. x - 3y = 0 5x - y = -14 12. 2x + 5y = 9 3x - 2y = 4 5
Solve the systems of equations using the addition method. Show all steps. 14. x + y = 12 13. 3x - 8y = 11 x + 6y - 8 = 0 1 2 1 x 4 4y 15. 2p q = 8 1 1 p q 3 3 4 Translate into a system of equations and solve using the addition method. 16. The sum of two numbers is 92. One eighth of the first 17. The difference of two numbers is 49. One half of one number plus one third of the second number is 19. Find number plus one seventh of the other number is 56. Find the the numbers. numbers. 18. The sum of two numbers is 115. The difference is 21. Find the numbers. 19. The sum of two numbers is 26.4. One is five times the other. Find the numbers. 20. Two angles are supplementary. One is 8 more than three times the other. Find the angles. (Supplementary angles are angles whose sum is 180 ) 21. Two angles are supplementary. One is 30 more than two times the other. Find the angles. 6
Roots and Radicals Practice I. The number c is a square root of a if c 2 = a. The square roots of 64 are 8 and -8. The principal square root of 64 is Written 64 = 8. The negative square root of 64 is written 64 = -8. Simplify 1. 36 2. 81 3. 49 4. 169 An irrational number cannot be named by fractional notation a. The rational numbers and the irrational numbers b make up the set of real numbers. Identify each square root as rational or irrational. 5. 3 6. 25 7. 12 8. 4 II. In a radical expression, the expression written under the radical symbol is called the radicand. Expressions with negative radicands have no meaning in the real number system. An example that shows how to simplify an 6 3 3 expression would be: 121y ( 11y )( 11y ) *The square root of a number is one of the two equal factors whose product = 11y 3 is that number. Simplify 9. m 2 10. p 4 11. 144x 2 12. 1 2 x 13. ( x 4) 2 4 III. For any nonnegative radicands, a and b, a b a b. You can use this property to multiply and factor radical expressons. A simplified radical expression has no factors which are perfect squares under the radical sign Multiply. Assume that all radicands are nonnegative. 14. 3 7 15. a t 16. x 3 x 3 17. 2x 3y 18. 3 4 5 19. 3x 2x 1 7 Factor and simplify. Assume that all radicands are nonnegative. 20. 48 21. 64x 2 2 22. x 14x 49 23. 36x IV. To find the square root of a power such as x 10,, the exponent must be even. Since (x 5 ) 2 = x 10 the exponent is odd, such as x 7, write the power as a product of an even and an odd power, (x 6. x). Then simplify the even power. = x = Assume that all radicals are nonnegative. Multiply and simplify. 24. 3 6 25. 2x 2 5x 5 26. ab bc 27. 5b 15b 3 7
Concept hints are included before each section. Read them thoroughly before working the problems. You should use a highlighter as you read to emphasize important facts, definitions or processes. GEOMETRY REVIEW SECTION