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Transcript
Mth 95 Notes Module 3 Review x2 – 6x – 7 ` Spring 2014 16 x 2 8 x 4 81n2 – 36 Section 5.8 Solving Equations by Factoring and Problem Solving Remember the quadratic formula from Mth 65… b b2 4ac If ax + bx + c = 0, then x = 2a We could use the quadratic formula. Solve x2 + 5x – 6 = 0 using factoring and solve x2 + 5x – 6 = 0. the zero-factor property ( If a and b are real numbers and a b 0 , then a 0 or b 0 ). 2 5 52 4(1)(6) x 2(1) 5 25 24 2 5 7 x 6,1 2 x Check by graphing in the standard window, and examine x-intercept or compute with 2nd trace (calc), zero(2) and follow the prompts. Since these are integer results, you could also examine the table. Why do we need both methods? Not all quadratic equations are factorable. Solve the following quadratic equations by factoring. Steps 1) Set equal to zero (standard form), if needed x2 – 2x – 35 = 0 2) Factor completely, if needed. 3) Set each factor equal to zero. 4) Solve resulting equations 5) Check each solution in the original equation (x + 2)(x – 5) = 0 2x(x + 3)(x – 1) = 0 x2 + 6x + 9 = 0 5x2 – 45 = 0 5x2 – 7x = -2 x2 + 4 = 8x – 12 Chapters 5 and 6 1 Mth 95 Notes Module 3 Spring 2014 If fractions are included in the equation, multiply by the LCM before you factor. x2 5 x 0 10 2 x2 x 1 0 18 2 You can use factoring to solve higher degree equations. x3 + 7x2 – 4x – 28 = 0 x4 – 26x2 + 25 = 0 Applications A rectangle is 6 inches longer than it is wide. Its area is 135 square inches. Find the dimensions of the rectangle. Chapters 5 and 6 2 Mth 95 Notes Module 3 Spring 2014 Remember the Pythagorean Theorem from Mth 65? The equation a b c refers to the relationship between the sides in a right triangle. The variables a and b are the lengths of the legs and the variable c is the length of the hypotenuse, which is the longest side of the right triangle. 2 2 2 The longer leg of a right triangle is 4 feet longer than the other leg. Find the lengths of the two legs if the hypotenuse is 20 feet. (Hint: Use the Pythagorean Theorem) The hypotenuse of a right triangle is 4 inches longer that the shortest leg. If the longer leg measures 8 inches, find the other dimensions of the triangle. Review: Solve each polynomial equation by factoring k2 + 5k = 0 2w2 = 5w – 3 Section 6.1 Rational Functions and Multiplying and Dividing Rational Expressions In section 5.1 we learned that MONOMIALS are terms in which the variables have ONLY NONNEGATIVE EXPONENTS and that POLYNOMIALS are either a monomial or a sum of monomials. Rational Expressions result when a polynomial is divided by a NONZERO polynomial. Examples: Chapters 5 and 6 3 Mth 95 Notes Module 3 Spring 2014 A rational function is given by f ( x) p( x) , where p(x) and q(x) are polynomials. The q( x) domain of f includes all x-values such that q(x) 0, that is, all real numbers where the ________________________________does not equal zero. Determining the DOMAIN of a rational function: The polynomial in the denominator CANNOT BE equal to ZERO because 3 dividing by zero is ______________________. If f(x) = , the denominator would be x zero when x =___. Therefore the domain of f(x) in set builder notation would be 3x3 15 x , the domain is 7 x2 5x 6 If g ( x) , the domain is x 2 25 Identify the domain of each rational function. Give your answer in set builder notation. x2 5x 6 x6 g(x) = h(x) = x4 x2 1 If h( x) GRAPHING a rational function 2 To graph f(x) = , make a table of values and plot the points. x x -4 -2 -1 -1/2 0 1/2 1 2 4 f(x) 2 does not cross x the line x = 0, the y-axis. A VERTICAL _________________ is a vertical line that typically occurs in the graph of a rational function when the denominator is 0 but the numerator is not 0. The equation for the vertical asymptote is ______________ Reducing Rational Expressions 12 4 Just as fractions can often be reduced, , rational expressions can be reduced. 15 5 Because f(0) is undefined, the graph of f(x) = x 2x 1 12 4 12 4 3 4 because . So, 2 x 6x 7 15 5 15 5 3 5 2 Remember you can only cancel factors, not terms! Chapters 5 and 6 x3 3. x 4 Mth 95 Notes Module 3 Spring 2014 Simplify each rational expression. Factor and cancel (remove the hidden factors of 1) y 2 4y 8 x2 4 x2 15 x 5 x3 x 2 7 x 18 x 2 3x 2 x3 4 x 2 3x 12 x4 Review Factor x3 3x 2 9 x 27 completely. Give the domain of Solve 2m2 7m 3 by factoring. x2 7 x 6 x2 2 x Simplify x 2 x 20 . 3 x 12 5x x 5 Multiplying Rational Expressions 1 4 1 2 2 2 Remember fractions: 2 3 23 3 Steps in multiplying rational expressions 1. Factor everything 5 x3 2 y 4 4 y 15 x x 2 1 x 2 x 2 2x x 1 2. Cancel, if you can 3. Write down what’s left Multiply the following rational expressions. x 3 x2 2x 8 x 4 x2 6 x 9 Chapters 5 and 6 5 Mth 95 Notes x 2 25 x2 2 x 3x 10 x Module 3 Spring 2014 x 2 6 x 9 4 x 12 2 x 2 18 5 x 15 Dividing Rational Expressions 2 x3 8 x 3 1 3 2 3 2 3 Remember fractions: 4 2 4 1 22 1 2 5 y 6 y2 Steps in dividing rational expressions 1. Flip the rational expression following the division sign (that gives you the reciprocal of the divisor) and replace the division symbol with a multiplication symbol. 2. Completely factor everything 3. Cancel what you can 4. Write down what’s left x 2 3x 10 x 2 5 x 6 2 2x x 3x 10n 15 6n 9 n2 1 n 1 4 40 y 6 42 7 y x 2 x 12 x 2 7 x 12 2 x2 9 x 5 2 x2 7 x 4 Simplify x 2 x 20 x 2 16 3 3x 15 x 4 x 5 Chapters 5 and 6 2 x 2 12 x x 2 3x 18 5 15 x 6 Mth 95 Notes Module 3 Spring 2014 Review Simplify x 2 x 20 . 3 x 12 Give the domain in set builder notation. 16w2 4w 3w2 14w 8 6.2 Additions and Subtraction of Rational Expressions Remember to add or subtract fractions, the fractions must have a common denominator ( the LCM of the denominators). 3 2 5 1 5 1 3 5 2 3 10 13 Ex1 Ex2 7 7 7 4 6 4 3 6 2 12 12 12 Steps in adding and subtracting rational expressions that have a common denominator. 1. Add or subtract the numerators 2. Factor everything, if you can 3. Simplify, if you can 2 x 5x 3y 3y 5y 5 y 1 y 1 x 2 x 4 3x 4 2 x2 4 x 4 y 3 5y 2 2y 2y Steps in adding and subtracting rational expressions that have unlike polynomial denominators. 1. Factor each polynomial in the denominator completely 2. List each factor the greatest number of times it occurs in each denominator. 3. The product of these factors is the LCD. 4. Rewrite each expression with the LCD by multiplying each fraction by one. 5. Add or subtract the numerators 6. Factor everything, if you can 7. Simplify, if you can Chapters 5 and 6 7 Mth 95 Notes 3 5 3 2 7x 2x Module 3 LCD = x 1 5 x x 5 LCD = Spring 2014 4 3 5 x 4 xy LCD = x x 1 2x 4 2x LCD = 1 1 2 2 x 3x 2 x x 2 x2 3 x 2 9 x 2 4x 3 x2 6 x4 2 x 9 x 18 x 6 x x 1 2 x 3 x 2x 3 Chapters 5 and 6 8