Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Positional notation wikipedia , lookup
Infinitesimal wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Big O notation wikipedia , lookup
Real number wikipedia , lookup
System of polynomial equations wikipedia , lookup
Vincent's theorem wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Prerequisite Chapter Homework Assignment Sheet Section P1: Review of Real Numbers and Their Properties Assignment 1: Pg 9 – 11 #’s 2 – 60 evens, 68 – 84 evens, 98 – 104 evens Section P2: Exponents and Radicals Assignment 2: Pg 21 – 22 #’s 2 – 36 even #’s Assignment 3: Pg 22 #’s 51 – 62 all Assignment 4: Pg 22 #’s 64 – 78 even #’s Assignment 5: Pg 22 – 23 #’s 80 – 102 even #’s Assignment 6: Pg 21 – 23 #’s 5 – 11, 25 – 35, 69 – 73, 91 – 101 ONLY ODD #’S Section P3: Polynomials and Special Products Assignment 7: Pg 29 - 30 #’s 2 – 6, 12 – 46, 56 - 60 Only even #’s Assignment 8: Pg 30 – 31 #’s 62 – 96 even #’s Plus 103 a and 105 Section P4: Factoring Polynomials Assignment 9: Pg 38 #’s 2 – 46 even #’s Assignment 10: Pg 38 #’s 51 – 64 all Assignment 11: Pg 38 – 39 #’s 65 – 78 all Assignment 12: Pg 39 #’s 79 – 111 odd #’s Section P5: Rational Expressions Assignment 13: Pg 48 #’s 2 – 28 even #’s Assignment 14: Pg 48 #’s 35 – 42 all Assignment 15: Pg 49 #’s 44 – 60 even #’s Assignment 16: Pg 48 – 49 #’s 7, 21, 25, 27, 43, 49, 51, 55, 57, 61 – 64 all, 67 Section P6: Errors and the Algebra of Calculus Assignment 17: Pg 56 #’s 1 – 18 all Assignment 18: Pg 56 – 57 #’s 19 – 22 all, 43 – 48 all, E.C. 60 Section P7: The Rectangular Coordinate System and Graphs Assignment 19: Pg 64 #’s 2 – 20 even #’s, 28, 30, 32 – 40 only b and c, 41, 48, 52 Review Prerequisite Chapter Assignment 20: Pg 70 – 71 #’s 2 – 66 evens except 18, 20, 32, 34 Assignment 21: Pg 71 – 72 #’s 67 – 86 all, 90, 92, 96, 99, 106, 108, 110, 112 2 P1: Review of real numbers and their properties A. Real numbers B. Ordering real numbers Ex. Put the following in order on the number line. 5 6 2 1 5, 4, 9.2, 9.05, 9.1, , ,.50, 4 , 4 9 13 5 7 Ex. Put the following on the number line. a. x 3 b. 7 x 6 3 There is another way of interpreting inequalities, describing them as __________ of real numbers called intervals. In bounded intervals like below, the real numbers a and b are the ______________ of each interval. Bounded intervals on the real number line Notation a, b Interval Type inequality a xb a, b a xb [a, b) a xb (a, b] a xb Graph Unbounded intervals on the real number line Notation [ a, ) Interval Type inequality xa a, xa ( , b] xb , b xb , x Ex. Graph Give a verbal description of the following, write it like an inequality and graph it a. 4,9 b. [3,8) c. (, 4] 4 C. Absolute Value and Distance The absolute value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. a, if a 0 Def: | a | a, if a 0 Properties of Absolute Values 1. | a | 0 2. | a || a | 3. | ab || a || b | Ex. 1. | 3 | | 9 | 3. |12 19 8 | 4. a a ,b 0 b b 2. 4. | 10 9 | | 7 29 || 29 7 | 5 P2. Exponents and Radicals Day 1 A. Integer Exponents Let’s complete the following table that shows the properties of exponents. Properties 1. x m x n Solution Ex 244 245 2. ( x m )n p 3. ( xy)m 3x 4. x m 62 5. x 0 100, 000,203, 0450 xm 6. n x 48 45 x 7. y 2 5 4 5 u v m 8. | a 2 | | 3 | 2 **Simplified means no no no no ( ) common bases decimals negative exponents unless told otherwise Examples: Evaluate each of the following using the rules of exponents. 1. (4)(4)3 2. [(3)2 ]3 3. (32 x 2 y 8 )2 4. 54 53 5. (23 )2 6. m 7 1 m4 6 7. 3 83 85 = 89 5 8. = 6 9. 5x 4 3x 3 y5 = 8 x5 6 y 4 2 10. 3 4 4 3 11. 4 4 2 3 12. 9 x 2 y 6 3 6 4x y Day 2 B. Radicals and their properties Principal nth Root of a Number: Let a be a real number that has at least one nth root. The principal nth root of a is the nth root that has the same sign as a . It is denoted by a radical symbol n a The positive integer n is the index of the radical, and the number a is the radicand. If n = 2, omit the index and write a rather than 2 a . ( The plural of index is indices). To simplify the following radicals with different indexes, you need to keep in mind such things as not only perfect squares, but also perfect cubes, etc of both positive and negative integers. 7 Ex. Simplify the following without using a calculator. 1. 49 2. 4. 3 27 5. 7. 3 64 8. 25 4 25 16 3. 16 6. 5 243 64 9. 4 81 3. 4 16 x 4 So, if a 0 and n is even, ____________. if a 0 and n is odd, _____________. Properties of Radicals Property 1. n a 2. n an b 3. Ex 3 3 a n b m n 5. a n 162 6 3 4 2 a n 27 4 32 2 n 4. 6. Solution m 7 n even: n xn n odd: n xn 17 127 x2 5 x5 Examples: 1. 2 18 4. 5 12 5 2. 3 27 x 3 5. 3 93 3 8 Day 3 C. Simplifying Radicals Radical in the simplest form must satisfy the following conditions: 1. All possible factors have been removed from the radical. 2. All fractions have radical-free denominators. (must rationalize the denominator) 3. The index of the radical is reduced. 1. D. 20 2. 50 3. 4. 3 64x5 5. 3 32x6 6. 7. 5 96x 5 y12 8. 4 32 9. 3 250 8x4 3 32x6 Combining Radicals Radicals can be combined only if they have the same index and radicand. You may have to simply the radical to eventually combine them. Ex. 1. 4 3 6 12 3 6 5 4 6 2. 5 3 2 3 16 8. 3 54 3 81 9. 4 32 2 4 16 3 4 162 9 Day 4 E. Rationalizing Denominators and Numerators When simplifying a rational expression, you cannot leave a radical in the denominator. You must eliminate the radical by multiplying the denominator by the conjugate of the form of a b m or a b m . Ex. 4 6 1. 2. 3 3 10 4 8 2 3 3. F. Rational Exponents 1 1 is the rational exponent of a. a n n a , Where n m 1n Additionally, a a m n Ex. a n m m n 1 3 5 2. 1 and a a m n n a m Rewrite in Radical notation 1. Ex. 12 5 2 4. Ex. Rewrite in exponent notation 1 6 12 5 3. 19 4. 9 28 Simplify the following rational exponents 1 1 1. 53 5 4 2. 1 4 3. 8 3 2 15 3 15 13 4. 2 2 10 5. 3 6. 125 x 3 x 3 2 3 P3: Polynomials and Special Products A. Polynomials To effectively manipulate polynomial expressions in solving real-life problems we must be able to add, subtract, multiply and divide the terms that make-up the polynomials and put your polynomial in standard form, which is writing the polynomial in descending order. Let’s classify the next 5 polynomials and identify the: _______________, ______________, ______________, ________________ and _________________. Polynomial 5 3x 2 4x 1 x 2 5x 3 8x 4 7x 2 11x 9 4x 9 Note: Notice polynomials involve only one variable with non-negative integer exponents. Is the following a polynomial: x 3 8x 2 4 9 ? ___________________ x When adding and subtracting polynomials remember to only combine ____________ terms. Ex 1: A. (5x 2 4x 8) (3x 2 2x 3) B. (8x 2 14x 3 4x 2) (6x 3 4x 2 7) 11 Ex 2: Multiplying by a monomial C. 4x 2 (6x 3 7x 5) Ex 3: Multiplying Polynomials (Horizontal v’s Vertical) D. (x 5)(2x 3 3x 2 4x 1) Horizontal B. Vertical Special Products Some binomial products occur so frequently that it is worth memorizing their special product patterns: Sum and Difference of same terms (u v )(u v ) u 2 v 2 Square of a binomial (u v )2 u 2 2uv v 2 (u v )2 u 2 2uv v 2 Cube of a binomial (u v )3 u 3 3u 2v 3uv 2 v 3 (u v )3 u 3 3u 2v 3uv 2 v 3 Ex 4: Recognizing Patterns A. (7 x 4)(7 x 4) B. (5 6x )2 C. (2x 9)3 D. (4x 7)3 12 C. Application Write a polynomial that expresses the area of the shaded region below 2x-1 x 3x 4x+3 P4: Factoring Polynomials A. Polynomials with common Factors When it comes to removing a common factor, we need to think back to Distributive property: a(b c) __________ . Now, let’s look at the Distributive property in reverse direction: ab ac ___________ . Notice the connection? Lets look at the following examples. Ex. Factor each expression by finding its common factor. 1. 8 y4 6 y 2. 9 x 3 54 x 27 3. B. 8 x 4 26 x 2 14 x 4. x 3 x x 3 9 Factoring Special Polynomials Forms Remember the special product forms we learned in P3, well here is the reverse form when it comes to factoring. 13 Difference of Two Squares: u 2 v 2 (u v )(u v ) 1. x 2 52 2. 3x 2 48 3. x 2 36y 2 4. 5 45x 2 5. x 6. 16 81x 4 2 5 16y 2 Perfect-square factoring: a. u 2 2uv v 2 b. u 2 2uv v 2 1. x 2 4x 4 2. x 2 6x 9 3. x 2 10x 25 4. x 2 8x 16 5. 16x 2 24x 9 6. 9x 2 42x 49 14. 3x 3 192 Sum of Two Cubes: u 3 v 3 (u v )(u 2 uv v 2 ) 13. x 3 27 15. 64x 3 27 y 3 Differences of 2 cubes: u 3 v 3 (u v )(u 2 uv v 2 ) 16. x 3 27 18. 8x 3 216 17. 125x 3 8 14 C. D. Trinomials with Binomial Factors There are many ways to factor trinomials. No matter what method you use, they are all guess and check. I personally use the box method but you can use any method you like. Ex. Factor the following trinomial when L.C. is 1 & using master product sum. 1. x 3 5x 2 6x Ex. Factor the following polynomials when L.C. is not 1. 4. 2x 2 7x 3 2. 5. 2x 2 16x 32 2x 2 x 3 3. 6. x 2y 2xy 1y 3x 2 11x 20 Factoring by Grouping Ex. Factor the following polynomial using grouping method. 7. 2x 3 3x 2 4x 6 Ex. 10. Factoring a trinomial using the grouping method. 2 x2 5x 3 11. 4 x 2 12 x 9 8. 5xy 10x 15y 30 9. xy 2x 3y 6 15 P5: Rational Expressions A. Domain of an Algebraic Expression By finding the domain, you are looking for the set of x-values that defines the expression. Ex. Which of the following functions have restrictions on their domains? a. f ( x) 4 x 9 Restriction: Restriction: c. h( x ) 2 x 6 l ( x) 6 x 18 d. Restriction: Restriction: e. m(t ) x3 x4 f. Restriction: n( x ) 5 4 x 32 Restriction: g. B. g ( x) 4 x 2 7 x 1 b. r ( x) 3x 4 9 Restriction: Simplifying Rational Expressions Reducing Rational Expressions: Factor the numerator and denominator first and then divide out common factors. Also denote what values x cannot equal to. 1. 8x 2y 24xy 5 2. 8x 3 2x 2 4x 2 x 3. x 2 49 x 2 8x 7 16 C. 1. 4. Operations with Rational Expressions Multiplying Rational Expressions: Factor first and then divide out common factors. 8x 3 14y 4 2x 2 x 3 x 2 x x 2 7x 8 4x 3 2. 3. 26xy 12x 2 x2 x 2x 3 3x 2 24x x 2 1 Dividing Rational Expressions: Change the division to multiplication of the reciprocal. 3x 1 x x 2 4x 4 x 1 5. x 2 6. (x 2 1) 3 3 2 2x 2x 2x 2x x 3 x 5 Adding and subtracting rational expressions - A. Find the least common denominator (LCD) Rewrite each rational expression using the LCD Add or subtract the numerator Reduce answer when possible 5 2 2 x 3x B. 3 4 x 5 x 1 17 C. D. A. C. x 3x 2 x 1 x 1 D. 5 3x 1 2 2 3x 12 x x 12 3 Complex Fractions and the Difference Quotient Simplifying Complex Fractions – Multiplying numerator and denominator by the LCD of every fraction. 1 2 x x 6 3 x 1 3 x B. 1 1 x 3 1 2 2x D. x2 2 x 1 3x x 1 18 E. Simplifying expressions with negative exponents 1. 5 3 3x 2 5 x (2 5 x) 2 3 2. x3 1 x 2 1 2 1 2 x 1 x 2 2 x4 P6: Errors and the Algebra of Calculus A. 1. Writing a fraction as a sum of terms x 4 9 x3 8x 2 12 x 18 2. 3x 8x 9 x 2 12 x 19 P7: The Rectangular Coordinate System and Graphs A. The Cartesian plane. Graphs of linear equations are geometric representations of the relationship between the variables. In this lesson we will use the coordinate plane to visualize this relationship between the variables. Ex. Coordinate Plane On the coordinate plane below, label each axis, each quadrant and origin. 10 Plot the following ordered pairs a) (3, 0), (-2, 0), (-10, 0), (8, 0) What kind of line is formed? b) What kind of line is formed? -10 -10 B. (0,8), (0,3), (0,-2), (0, - 10) 10 The Distance Formula: It is derived from the Pythagorean Theorem. ( x1 , y1 ) and ( x2 , y2 ) is d . 20 C. Ex. Find the distance between (-2, 12) and (5, 7). Ex. Verify that the following three ordered pairs creates a right Triangle. 2,1 , 4,0 , 5,7 The Midpoint Formula: It is used to find the Mid-point between two points x1, y1 & x2 , y2 . x x y y2 Midpoint 1 2 , 1 2 2 Ex. Find the midpoint between the following segment: 4, 8 & 1,10 21