ࡱ> ~'` RbjbjDD >&&pJ&8(z "      $h       Ӄ    o|x  G-r  rN0s^~|x|x`       (X       ̙$j̙ Math 72 Course Pack Fall 2010 Section 0249 ONLY Instructor: Yolande Petersen DO NOT BUY THESE NOTES IF YOU HAVE A DIFFERENT INSTRUCTOR Inside: Lecture Notes Outline with writing space for your notes Syllabus Calendar of Material Coverage and Exam Dates Homework Assignments for each section Cumulative Review Problems (sample test problems, Petersen style only) Final Exam Review Suggestions for use The lecture notes pages are in reverse order, upside down, and punched on the "wrong" side for a reason! My notes read like a book, with the printed side on the left and handwritten extra notes on the right. If you want your notes look like mine, insert the lecture pages into a binder with the blank back of page 1 on top. When you turn the first page, page 1 will be on your left, with the blank back of page 2 on your right, where you can write additional notes. Effort was made to minimize the number of pages printed to reduce your cost, while leaving enough space for your notes to be arranged in an orderly way. If you don't like this arrangement, feel free to order the pages however you like. The supplements (syllabus, homework assignments, and review problems) can be inserted into a binder in the "usual" way. Mrs. Petersen's website: http://www.virtual.mjc.edu/peterseny Before you take this class, you may find it helpful to read the document at the link, "Teaching Style and Educational Philosophy" to decide whether this instructor is a good match for you. Chapter 5 Review Concepts Distributive Property (multiplying polynomials by monomials) Ex a FOIL (multiplying 2 binomials) First, Outer, Inner, Last Special Product Formulas (f + s)2 = f2 + 2fs + s2 (f s)2 = f2 2fs + s2 (f + s)(f s) = f2 s2 Common Errors: 6.1 The Greatest Common Factor (GCF); Factoring by Grouping Factors - things multiplied to make a product It is sometimes desirable to write things as factors (pieces) Prime factored form - a product of factors where none of the factors can be "broken down" further (with exponents consolidated) Greatest Common Factor (GCF) the largest factor (small broken-down piece) included in all the numbers Least Common Multiple (LCM) - the smallest multiple (large multiplier) that includes all the numbers Finding the GCF Write all the numbers in prime factored form For each factor, choose the smallest exponent common to all the numbers Write the productFinding the LCM Write all the numbers in prime factored form For each factor choose the largest exponent possible Write the product Factoring out the GCF the reverse of the distributive law Find the GCF (by eyeballing or previous method) and write it in front Divide the GCF out of each term Write the "leftovers" inside parentheses Factoring by Grouping take out identical "clumps" of stuff 6.2 Factoring Trinomials (of form x2 + bx + c) reverse of FOIL x2 + bx + c factors to (x + ?)(x + ?) How do we know whether it's + or - , and how do we get ? Some examples of FOIL Factored Unfactored (x + 1)(x + 2) = x2 + x + 2x + 2 = x2 + 3x + 2 (x 3)(x 5) = x2 5x 3x + 15 = x2 8x + 15 (x 4)(x + 3) = x2 + 3x 4x 12 = x2 x 12 (x 2)(x + 5) = x2 + 5x 2x 10 = x2 + 3x 10 Observations on signs If c is positive  EMBED Equation.3  If b is positive  EMBED Equation.3  If b is negative  EMBED Equation.3  If c is negative  EMBED Equation.3  If b is positive  EMBED Equation.3  If b is negative  EMBED Equation.3  Observations on numbers c is the product of the 2 binomial numbers  EMBED Equation.3  If the 2 binomial numbers have the same sign, b If the 2 binomial numbers have different signs, b 6.3 More Factoring Trinomials (ax2 + bx + c) ax2 + bx + c factors to (?x + ?)(?x + ?) Factoring by grouping long process, no guesswork Multiply EMBED Equation.3  For c positive, find 2 factors whose sum = b For c negative, find 2 factors whose difference = b Rewrite the whole polynomial with the middle term written as the sum or diff. Factor by grouping Ex a Trial and Error For simple numbers with few factors, it's much faster than grouping For numbers with many factors, requires much guesswork and luck Rules for signs are same as section 6.2 Rules for numbers require "mix and match" to get the middle number, b 6.4 Special Factoring Special formulas (same as before, written in reverse): f2 s2 = (f + s)(f s) f2 + 2fs + s2 = (f + s)2 f2 2fs + s2 = (f s)2 Ask: Does my binomial or trinomial fit one of these 3 patterns Binomials Use 1st formula for squares, last 2 formulas for cubes Trinomials - Use formulas 2 or 3 More Formulas cubes f3 + s3 = (f + s)(f2 fs + s2) f3 s3 = (f s)(f2 + fs + s2) Common Errors: x3 + y3  EMBED Equation.3  (x + y)3 x3 y3  EMBED Equation.3  (x y)3 Note: x2 y2 can be factored x2 + y2 can't be factored Factoring Summary Can a GCF be factored out? (6.1) 2 terms: Is it (f2 s2) or (f3 + s3) or (f3 s3)? (6.4) 3 terms: Is it (f2 + 2fs + s2) or (f2 2fs + s2)? (6.4) Does trinomial  EMBED Equation.3  2 binomial factoring work easily? (6.2) Is there a number in front of x2 for trinomial  EMBED Equation.3  2 binomials? (6.3) 4 terms: Does factoring by grouping work? Can any factors be further factored? 6.5 Solving Quadratic Equations Factoring Quadratic equation 2nd degree Standard form: ax2 + bx + c = 0 Zero Factor Property basis for solving quadratic equations If ab = 0, then a = 0 or b = 0 (one of the numbers MUST be zero) Ex a Procedure for solving Write equation in standard form (get 0 on one side) Factor Set each piece (factor) equal to 0 Solve each piece; check The number of solutions is equal to or less than the degree of the equation. How what is the maximum number of solutions of a quadratic equation? How many solutions could a cubic equation potentially have? 6.6 Applications of Quadratic Equations Rectangles: A = LW; P = 2L + 2W Triangles: Pythagorean Theorem: In a right triangle with hypotenuse c and legs a & b, a2 + b2 = c2 Ex The hypotenuse of a right triangle is 1 cm longer than the longer leg of the triangle. The shorter leg is 7 cm shorter than the longer leg. Find the length of the longer leg. Numbers Velocity/Distance Ex The height of a ball with an initial velocity of 128 ft/sec which travels t seconds is described by the equation: h = 128t 16t2 What is the height of the ball after 2 seconds? For what values of t is h = 0? Interpret the values from b) in real life terms. For what values of t is h = 240 ft? After how many seconds do you think the ball will reach its peak? 7.1 The Fundamental Property of Rational Expressions (Canceling Fractions) Rational Expression: Can be written as P/Q, where P and Q are polynomials , Q EMBED Equation.3 0 (Note: Q = 1, so non-fractions can be made into fractions) P/Q is undefined when Q = 0. A place where the denominator is undefined is called a Ex a For what values is the expression  EMBED Equation.3  undefined? Lowest Terms The rational expression P/Q is in lowest terms if there are no common factors in the denominator. Fundamental Property of Rational Expressions (Canceling Rule)  EMBED Equation.3  where K  EMBED Equation.3  0 7.2 Multiplying and Dividing Rational Expressions Multiplying:  EMBED Equation.3  Smart Way: Cancel (reduce) as much as possible before multiplying Procedure: Factor Cancel and rewrite the remaining factors Note which bad points have disappeared (for extra credit) Dividing:  EMBED Equation.3  7.3 The Least Common Denominator (LCD) Recall: The LCD is the least common multiple of the denominators. It is as large as or larger than each of the individual denominators. Finding the LCD of Polynomial Factors Factor each denominator completely, writing repeat factors in exponent form Write each unique factor to the highest power possible Caution: Don't confuse the LCD with the GCF. Ex a Building a fraction to match a denominator Compare the new and old denominators, and find "what's missing" from the old denominator. When in doubt, divide the LCD by the old denominator. Multiply top and bottom by the missing factor. 7.4 Adding and Subtracting Rational Expressions Rules:  EMBED Equation.3  and  EMBED Equation.3  for Q EMBED Equation.3 0 Caution: 2 fractions must have same denominator. If not, a common denominator (LCD) must be used. Procedure -- Adding/Subtracting with same denominator Add or subtract numerators, keep denominator Factor and/or cancel if possible Adding/Subtracting with different denominators Find the LCD Build the fractions to match LCD Add or subtract as above 7.5 Complex Fractions Typically have 4 layers (2 top, 2 bottom) We can treat 4 layers as 2 separate fractions, divided 2 Solving methods Method 1: Invert bottom fraction and multiply Method 2: Multiply both fractions by LCD to clear 2 denominators Method 1 is generally more reliable; Method 2 can be fast but is prone to errors Clearly Separated Layers Use Method 1 Ex a Partial Layers Use either method 7.6 Equations with Rational Expressions Recall: ExpressionsEquationsno equal sign Goal: Keep denominator by putting numerator over LCD Final answer may have mixed variables and numbers Process is to simplifyhas equal sign Goal: get rid of denominator by multiply by LCD Final answer: x = number (variable is isolated) Process is to solve  Solving Procedure Find LCD Multiply both sides by LCD Cancel factors to get rid of denominators Solve ***Check if solution is a bad point This is now required, not optional Solving for a Specified Variable Clear denominators if necessary Get all terms with desired variable on one side, all other terms on other side Factor out desired variable Divide by "junk" 7.7 Applications Distance/Rate/Time d = rt; isolating for different variables, r = d/t or t = d/r Work/Rate/Time w = rt or r = w/t or t = w/r "3 hours to paint a room" can be interpreted in 2 ways: t = 3 hours r = 1 room/3 hours or 1/3 room/hour (this is most commonly used) 8.1 Evaluating Roots square rooting - the reverse process of squaring Suppose we are given x2 = 25. What is the "unsquare" of 25? All square roots of "a" If "a" is a positive, real number,  EMBED Equation.3  is the positive square root of "a" -  EMBED Equation.3  is the negative square root of "a" Note the difference between word form and symbol form: "the square roots of x"  EMBED Equation.3  Types of Roots Rational Roots If "a" is a perfect square, then  EMBED Equation.3  is rational, e.g. Irrational Roots If "a" is positive but not a perfect square,  EMBED Equation.3  is irrational To get a ballpark idea of a number: Non-real roots If "a" is negative,  EMBED Equation.3  is non-real Pythagorean theorem c2 = a2 + b2 or  EMBED Equation.3  Higher roots It's helpful to know some perfect squares, cubes, 4th powers, etc. x123456789101112131415x2149162536496481100121144169196225x3182764125216x411681256625x5132243 Example:  EMBED Equation.3  "The cube root of 8" 8.2 Multiplying, Dividing, Simplifying Radicals Multiplying: Product Rule  EMBED Equation.3  and  EMBED Equation.3  Simplifying: A radical is completely simplified when no perfect square factor is inside the radical symbol. 2 Methods for simplifying: Method 1 Find prime factorization & take out even powers Method 2 "Eyeball" perfect squares that are easy to see & take them out until no perfect squares are left Quotient Rule  EMBED Equation.3  and  EMBED Equation.3  Radicals with variables For even powers inside square roots divide power by 2 For odd powers inside square roots separate off 1 power, divide the rest by 2 8.3 Adding & Subtracting Radicals Don't confuse with multiplication:  EMBED Equation.3  Radicals are similar to variable like terms only like types can be added/subtracted Higher roots Only like radicals (same index and number under radical) can be combined Goal: For cube roots, take out perfect cubes, e.g 8, 27, 64, 125 or x3 x6, x9 For 4th roots, take out perfect 4th powers, e.g. 16, 81, 256 or x4, x8, x12 8.4 Rationalizing the Denominator Completely simplified radicals have: No perfect squares inside radical sign No fractions inside radical sign No radicals in the denominator Procedure for Rationalizing the Denominator Simplify the denominator radical as much as possible Examine the what remains inside the radical and determine "what's missing" to make a whole (non-radical) Multiply top and bottom by the missing part 8.5 Simplifying Radical Expressions Completely simplified radicals should have: No parentheses (distribute or multiply out all terms) No perfect square factors inside radical No fractions inside radical or radicals in the denominator Products of radicals should be written under one "roof" Sums of like radicals should be combined Multiplying distributive law Multiplying FOIL Conjugates a pair of binomials that have 1) 1st terms exactly the same and 2) 2nd terms that are the same, except for opposite signs Writing a radical in lowest terms You CAN'T cancel terms (things added/subtracted) in the top & bottom You CAN cancel factors (things multiplied) in top & bottom Method 1: Factor top and bottom, then cancel Method 2: Separate numerator terms, putting each over its own denominator, then cancel 8.6 Solving Equations with Radicals Goal: Isolate the variable Recall: Squaring is the reverse of square root, so squaring a radical "undoes" it Procedure: Isolate the radical. If there are 2 radicals, get one on each side Square both sides. Combine like terms If there is still a radical, isolate it. Repeat steps 1 & 2. Solve for potential solutions Check all potential solutions this is mandatory! 8.7 Fractional Exponents By definition: a =  EMBED Equation.3  a 1/3 =  EMBED Equation.3  a 1/n =  EMBED Equation.3  To verify: Other fractional Exponents a m/n =  EMBED Equation.3  a m/n = (am)1/n = (a 1/n)m Observation on solutions: Linear (first degree) equations typically have 1 solution Quadratic (2nd degree) equations typically have 2 solutions Radical equations (1/2 power) typically have solutions that are good "half the time" Fractional Expressions Recall: Product rule:  EMBED Equation.3  Quotient rule:  EMBED Equation.3  Power rule:  EMBED Equation.3  9.1 Solving Quadratic Equations Square Root Property Recall: Quadratic equations look like ax2 + bx + c = 0 (2 solutions maximum) Ex a Solve x2 5x = - 4 4 Common Methods of Solving Quadratic Equations MethodAdvantagesDisadvantages1. Factoringfast, simpleDoesn't solve every equation2. Square Rootfast, simpleDoesn't solve every equation3. Complete the SquareSolves every equationRequires thinking4. Quadratic FormulaSolves every equationTedious, requires many steps Square Root Method works when b 0 (only have x2 and constant terms, no x term) Square Root Property If a2 = k, then a = +  EMBED Equation.3  or a =  EMBED Equation.3  Remember: Quadratics may have 2 solutions  EMBED Equation.3  x2 = positive number (or (stuff)2 = positive #) no solution  EMBED Equation.3  x2 = negative number one solution  EMBED Equation.3  x2 = 0 9.2 Completing the Square Suppose we have x2 + 10x + 25 = 36 Procedure for Completing the Square Get equation in the form x2 + bx = c (x2 and x terms on left, number on right) Find the needed number to complete the square nn =  EMBED Equation.3  Add it to both sides. Factor the perfect square. Write it as a square that looks like (x + b/2)2 Square root both sides Solve for x 9.3 Quadratic Formula Recall: Quadratic equations have the form ax2 + bx + c = 0 Quadratic equations have 2, 1, or 0 solutions Quadratic Formula The equation ax2 + bx + c = 0 has solutions: x =  EMBED Equation.3  or x =  EMBED Equation.3  Putting the 2 together: x1,2 =  EMBED Equation.3  Your book derives this formula on p. 608. We won't! The inside of the square root is sometimes called the discriminant. If b2 4ac > 0  EMBED Equation.3  2 real roots If b2 4ac = 0  EMBED Equation.3  1 real root If b2 4ac < 0  EMBED Equation.3  no real root 9.4 Complex Numbers The number i is defined as i =  EMBED Equation.3  (a non-real number) i2 = -1 Complex numbers have the form a + bi e.g. -3 + 2i or 7 - 4i There are pure real, pure imaginary numbers, and numbers with a mix of each. Imaginary numbers refers to any number with an imaginary part (either pure imaginary or a mix of real & imaginary parts). Complex numbers include all 3 types (pure real, pure imaginary, mixed) Adding and Subtracting add/subtract real and imaginary parts separately, similar to like terms Dividing Imaginary numbers are not allowed in the denominator Rationalize by multiplying top and bottom by the conjugate, similar to the way radicals in the denominator are eliminated. Multiplying imaginary conjugates produces a real number: (a + bi)(a - bi) = a2 abi + abi b2i2 = a2 b2(-1) = a2 + b2 9.5 Graphing Quadratic Equations/Quadratic Functions Recall some features of linear functions (1st degree): They can be written as y = mx + b They are straight lines Important landmarks are m (slope) and b (y-intercept) Some features of quadratic equations (2nd degree): They can be written as y = ax2 + bx + c They are Important landmarks include: Direction of opening Vertex and axis of symmetry Width (of parabola) Finding the Vertex (better way than trial & error) x-coordinate: x =  EMBED Equation.3  EMBED Equation.3  y-coordinate: plug the value of x into the original equation to get y Procedure for Graphing a 2nd degree equation (parabola) Decide if the parabola opens up or down Calculate the vertex Plot at least 1 extra point to find the width of the parabola Intercepts of a Graph of a Quadratic Function A graph has its x-intercepts when y = f(x) = 0 (set equation = 0) The x-values are the places where the graph crosses the line.      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