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Math 90 14.1 Multiplying and Simplifying Rational Expressions A. Rational Expressions and Replacements P * A rational expression is an expression that can be written in the form ; where P Q and Q are polynomials. x4 Ex. Find the numerical value of when x 3 . 2x 2 4 2 2 * Because rational expressions indicate division, we must be careful to avoid denominators of zero. When a variable is replaced with a number that produces a denominator equal to zero, the rational expression is not defined. Ex. Find all numbers for which the given rational expression is not defined. x4 x 2 3 x 10 *Note* The value of the numerator has no bearing on whether or not a rational expression is defined. To determine which numbers make the rational expression not defined, we set the denominator equal to 0 and solve. B. Multiplying by 1 * We multiply rational expressions in the same way that we multiply fraction notation in arithmetic. Multiplying Rational Expressions: To multiply rational expressions, multiply numerators and multiply denominators. A C AC B D BD For example, x2 x2 x 2 x 2 3 x7 3 x 7 M. Ruvalcaba Math 90 * Any rational expression with the same numerator and denominator is a symbol for 1. 19 x2 3x 2 4 1, 1, 1 19 x2 3x 2 4 Ex. 3x 2 1 3x 2 2x 3x 2 2 x x 1 x 1 2x x 1 2 x Ex. x2 x3 x 2 x 3 x7 x3 x 7 x 3 Ex. 2 x 1 2 x 1 2 x 1 2 x 1 C. Simplifying Rational Expressions Ex. Simplify each rational expression. ( by factoring and reducing ) 15 35 3 1. 20 225 4 8x2 8 x x x 2. 24 x 38 x 3 3. x2 9 x 3 x 3 x3 x2 x 6 x 3 x 2 x2 5x 5 4. x3 x2 M. Ruvalcaba Math 90 x2 8x 7 5. x2 4x 5 x2 4x 4 6. x2 2x x4 x3 7. 5x 5 x 2 11 x 18 8. x2 x 2 x 2 10 x 25 9. x2 5x x7 10. x 2 49 x7 11. 7x M. Ruvalcaba Math 90 D. Multiplying and Simplifying Ex. Multiply and simplify. x 3 2 x 10 12. x 5 x2 9 7 x 2 3 y 5 13. 5y 14 x 2 x2 x 6 14. 3x 5x 5 3x 3 2x2 x 3 15. 5x2 5x 4x2 9 x2 6x 9 x 2 16. x2 4 x3 M. Ruvalcaba Math 90 14.2 Division and Reciprocals A. Finding Reciprocals * Two expressions are reciprocals of each other if their product is 1. The reciprocal of a rational expression is found by interchanging the numerator and the denominator. 2 5 2 5 1. The reciprocal of is . (Note: 1) 5 2 5 2 2x2 3 x4 2x2 3 x4 2. The reciprocal of is . (Note: 1) x4 2x 3 2 x4 2x2 3 B. Division * We divide rational expressions in the same way that we divide fraction notation in arithmetic. Dividing Rational Expressions: To divide by a rational expression, multiply by its reciprocal and then factor and, if possible, simplify. A C A D AD B D B C BC Ex. Divide and simplify. 2 3 1. x x 3x3 4x3 2. 2 40 y 4 28 x 3. 2 x 7x 2 x 49 M. Ruvalcaba Math 90 6x 2 3x 2 x 4. x2 1 x 1 x 1 x 1 5. 2 x 1 x 2x 1 2 2 x 2 11 x 5 4x 2 6. 5 x 25 10 x2 2x 3 x 1 7. x2 4 x5 M. Ruvalcaba Math 90 14.6 Solving Rational Equations A. Rational Equations * In Sections 14.1 and 14.2, we studied operations with rational expressions. These expressions have no equal signs. We can multiply, divide, and/or simplify expressions, but we cannot solve if there are no equal signs. Most often, the result of our previous calculations is another rational expression that has not been cleared of fractions. Equations, on the other hand, do have equals signs, and we can clear them of fractions. A rational equation is an equation containing one or more rational expressions. Solving Rational Equations: To solve a rational equation, the first step is to clear the equation of fractions. To do this, multiply all terms on both sides of the equation by the LCM of all the denominators in the equation. Then carry out the equation-solving process. Finding Least Common Multiples Ex. Find the LCM of the denominators of each pair of rational expressions. 1 3 7 6 , , 8 22 5x 15 x 2 8222 5x 5 x 22 2 11 15 x 2 3 5 x x LCM 2 2 2 11 88 LCM 3 5 x x 15 x 2 7x 5x2 6m 2 2 , , x2 x2 3 m 15 m 5 2 x2 3 m 15 x2 m 5 2 LCM LCM M. Ruvalcaba Math 90 t 10 t5 y5 y4 , , t t 6 2 t 3t 2 2 y 2y 3 2 y 3y 2 2 t2 t 6 y2 2y 3 t 2 3t 2 y2 3y 2 LCM LCM 2 5 x Ex. Solve: *Note* The LCM of all denominators is 2 3 3 , or 18. We 3 6 9 multiply all terms on both sides by 18. 2 5 x 18 18 3 6 9 2 5 x 18 18 18 3 6 9 12 15 2 x 27 2 x 27 x 2 x x 1 x x 1 Ex. Solve: Ex. Solve: 6 8 12 4 6 8 M. Ruvalcaba Math 90 * If any denominator in the original equation contains a variable, you must be sure that the proposed solution does not make any of the rational expressions, in the original equation, undefined. That is, the solution cannot make any denominator 0 in the original equation. Check the solution! 1 1 Ex. Solve: *Note* The LCM of all denominators is x 4 x . We multiply x 4x all terms on both sides by x 4 x . 1 1 x 4 x x 4 x x 4x 4xx 4 2x x2 1 1 2 1 Ex. Solve: Ex. Solve: 10 x 6x 3x x 6 1 Ex. Solve: x 5 Ex. Solve: x 2 x x M. Ruvalcaba Math 90 x2 4 x2 1 Ex. Solve: Ex. Solve: x2 x2 x 1 x 1 4 1 26 3 1 2 Ex. Solve: 2 Ex. Solve: 2 x2 x2 x 4 x5 x5 x 25 M. Ruvalcaba Math 90 14.7 Applications Using Rational Equations and Proportions A. Solving Applied Problems Problems Involving Work: Ex. Erin and Tara work as volunteers at a community recycling depot. Erin can sort a morning‟s accumulation of recyclables in 4 hr, while Tara requires 6 hr to do the same job. How long would it take them, working together, to sort the recyclables? Ex. By checking work records, a contractor finds that it takes Eduardo 6 hr to construct a wall of a certain size. It takes Yolanda 8 hr to construct the same wall. How long would it take if they worked together? M. Ruvalcaba Math 90 Problems Involving Motion: * Problems that deal with distance, speed (or rate), and time are called motion problems. Translation of these problems involves the distance formula, d r t . Ex. A zebra can run 15 mph faster than an elephant. A zebra can run 8 mi in the same time that an elephant can run 5 mi. Find the speed of each animal. Ex. Nancy drives 20 mph faster than her father, Greg. In the same time that Nancy travels 180 mi, her father travels 120 mi. Find their speeds. M. Ruvalcaba Math 90 B. Applications Involving Proportions A C * An equality of ratios, , is called a proportion. The numbers within a B D proportion are said to be proportional to each other. Ex. A 2004 Toyota Prius is a gasoline-electric car that travels 240 mi in city driving on 4 gal of gas. Find the amount of gas required for 360 mi of city driving. Ex. To determine the number of fish in a lake, a park ranger catches 225 fish, tags them, and throws them back into the lake. Later, 108 fish are caught, and 15 of them are found to be tagged. Estimate how many fish are in the lake. Similar Triangles: * In similar triangles, corresponding angles have the same measure and the lengths of corresponding sides are proportional. Ex. Triangles ABC and XYZ are similar triangles. Solve for z. Y B 5 8 z 10 A C X Z M. Ruvalcaba Math 90 16.1 Introduction to Radical Expressions A. Square Roots * When we raise a number to the second power, we have squared the number. Sometimes we may need to find the number that was squared. We call this process finding the square root of a number. Square Root: The number c is a square root of a if c 2 a . * Every positive number has two square roots. For example, the square roots of 25 are 5 and -5 because 5 2 25 and 5 25 . The positive square root is also called the 2 principal square root. The symbol is called a radical symbol. The radical symbol represents only the principal square root. Thus, 25 5 . To name the negative square root of a number, we use . The number 0 has only one square root, 0. Ex. Find the square roots of 81. The square roots are 9 and - 9. Ex. Find 225 . There are two square roots of 225, 15 and - 15. We want the principal, or positive, square root since this is what represents. Thus, 225 15 . Ex. Find 64 . The symbol 64 represents the positive square root. Then 64 represents the negative square root. That is, 64 8 , so 64 8 . C. Applications of Square Roots Ex. After an accident, how do police determine the speed at which the car had been traveling? The formula r 2 5 L can be used to approximate the speed r, in miles per hour, of a car that has left a skid mark of length L, in feet. What was the speed of a car that left skid marks of length (a) 30 ft? (b) 150 ft? M. Ruvalcaba Math 90 a) We substitute 30 for L and find an approximation: r 2 5 L 2 5 30 2 150 24.495 . The speed of the car was about 24.5 mph. b) We substitute 150 for L and find an approximation: r 2 5 L 2 5 150 54.772 The speed of the car was about 54.8 mph. D. Radicands and Radical Expressions * When an expression is written under a radical, we have a radical expression. The expression written under the radical is called the radicand. E. Expressions That Are Meaningful as Real Numbers * The square of any nonzero number is always positive. There are no real numbers that when squared yield negative numbers. Thus the following expressions do not represent real numbers (they are meaningless as real numbers): 100 , 49 , 3 Excluding Negative Radicands: Radical expressions with negative radicands do not represent real numbers. F. Perfect-Square Radicands * In general, when replacements for x are considered to be any real numbers, it follows that x2 x , and when x 3 or x 3 , x2 32 3 3 x2 3 3 3 2 and M. Ruvalcaba Math 90 Principle Square Root of A 2 : For any real number A , A 2 A . (That is, for any real number A , the principal square root of A 2 is the absolute value of A .) Ex. Simplify. Assume that expressions under the radicals represent any real number. 1. 10 2 10 10 7 7 7 2 2. 3x 3x 2 3. a2b2 ab ab 2 4. x2 2x 1 x 1 x 1 2 5. * Fortunately, in many cases, it can be assumed that radicands that are variable expressions do not represent the square of a negative number. When this assumption is made, the need for absolute-value symbols disappears. Principal Square Root of A 2 : For any nonnegative real number A , A 2 A . (That is, for any nonnegative real number A , the principal square root of A 2 is A .) Ex. Simplify. Assume that radicands do not represent the square of a negative number. 3x 3x 2 6. a2b2 ab ab 2 7. x2 2x 1 x 1 x 1 2 8. Radicals and Absolute Value: Henceforth, in this text we will assume that no radicands are formed by raising negative quantities to even powers. M. Ruvalcaba Math 90 16.2 Multiplying and Simplifying with Radical Expressions A. Simplifying by Factoring * To see how to multiply with radical notation, consider the following. a) 9 4 3 2 6 b) 94 36 6 * Note that 9 4 94 The Product Rule for Radicals: For any nonnegative radicands A and B , A B A B . (The product of square roots is equal to the square root of the product of the radicands.) Ex. Multiply. 1. 5 7 57 2. 8 8 88 2 4 2 4 3. 3 5 3 5 4. 2x 3x 1 2x 3x 1 * To factor radical expressions, we can use the product rule for radicals in reverse. Factoring Radical Expressions: A B A B * In some cases, we can simplify after factoring. When simplifying a square-root radical expression, we first determine whether the radicand is a perfect square. Then we determine whether it has perfect-square factors. The radicand is then factored and the radical expression simplified using the preceding rule. M. Ruvalcaba Math 90 Compare the following: 50 10 5 10 5 50 25 2 25 2 5 2 * In the second case, the radicand has the perfect-square factor 25. Thus, it is simplified. Square-root radical expressions in which the radicand has no perfect-square factors, such as 5 2 , are considered to be in simplest form. Ex. Simplify by factoring. 5. 18 92 9 2 6. 48 t 16 3 t 7. 20 t 2 45t2 x2 6x 9 x 3 2 8. 9. 36 x 2 36 x2 3x 2 6 x 3 3 x2 2x 1 3 x 1 2 10. B. Simplifying Square Roots of Powers * To take the square root of an even power such as x 10 , we note that x 10 x 5 . Then 2 x 2 x 10 5 x 5 . We can find the answer by taking half the exponent. That is, x 10 x 5 . Ex. Simplify. 11. x6 x3 M. Ruvalcaba Math 90 12. x8 x4 13. t 22 * If an odd power occurs, we express the power in terms of the largest even power. Then we simplify the even power as in Examples 11-13. Ex. Simplify by factoring. 14. x9 x8 x 15. 32 x 15 16 2 x 14 x 16. 24 x 11 C. Multiplying and Simplifying * Sometimes we can simplify after multiplying. We leave the radicand in factored form and factor further to determine perfect-square factors. Then we simplify the perfect- square factors. Ex. Multiply and then simplify by factoring. 17. 2 14 2 14 28 47 2 7 18. 3 6 19. 2 50 20. 3x 2 9x3 3x 2 9 x3 27 x 5 9 3 x 4 x 3x 2 3x 21. 2x3 8x3 y 4 22. 20 c d 2 35 c d 5 M. Ruvalcaba Math 90 16.3 Quotients Involving Radical Expressions A. Dividing Radical Expressions * To see how to divide with radical notation, consider the following. 25 5 a) 16 4 25 5 b) 16 4 25 25 * Note that 16 16 The Quotient Rule for Radicals: For any nonnegative number A and any positive A A number B, . (The quotient of two B B square roots is equal to the square root of the quotient of the radicands.) Ex. Divide and simplify. 27 27 1. 9 3 3 3 96 2. 6 75 3. 3 30 a 5 30 a 5 4. 5a 3 5 a2 a a 5a 6a 2 6a 2 M. Ruvalcaba Math 90 42 x 5 5. 7x2 B. Square Roots of Quotients * To find the square root of certain quotients, we can reverse the quotient rule for radicals. We can take the square root of a quotient by taking the square roots of the numerator and the denominator separately. Square Roots of Quotients: For any nonnegative number A and any positive number B, A A . (We can take the square roots of the B B numerator and the denominator separately.) Ex. Simplify by taking the square roots of the numerator and denominator separately. 25 25 5 6. 9 9 3 49 7. t2 * Sometimes a rational expression can be simplified to one that has a perfect-square numerator and a perfect-square denominator. Ex. Simplify. 18 92 9 3 8. 50 25 2 25 5 18 2 9. 32 2 M. Ruvalcaba Math 90 48 x 3 48 x 3 16 4 10. 2 3x7 3x 7 x4 x 98 y 11. 11 2y C. Rationalizing Denominators * Sometimes in mathematics it is useful to find an equivalent expression without a radical in the denominator. This provides a standard notation for expressing results. The procedure for finding such an expression is called rationalizing the denominator. Ex. Rationalize the denominator. 2 2 2 3 6 6 12. 3 3 3 3 9 3 3 3 3 13. 5 5 5 5 14. 18 8 15. 7 3 16. 2 5 17. x 49 a 5 18. 12 M. Ruvalcaba Math 90 16.4 Addition, Subtraction, and More Multiplication A. Addition and Subtraction * We can add any two real numbers. The sum of 5 and 2 can be expressed as 5 2 . We cannot simplify this unless we use rational approximations such as 5 2 5 1.414 6.414 . However, when we have like radicals, a sum can be simplified using the distributive laws and collecting like terms. Like radicals have the same radicands. Ex. Add or subtract. Simplify, if possible, by collecting like radicals. 1. 3 5 4 5 3 4 5 7 5 2. 8 5 3 5 3. 5 2 18 5 2 92 5 2 3 2 2 2 4. 2 10 7 40 5. 24 54 6. 4x3 7 x 7. x3 x2 4x 4 M. Ruvalcaba Math 90 1 8. 3 3 B. Multiplication * Now let‟s multiply where some of the expressions may contain more than one term. Ex. Multiply. 9. 2 3 7 6 14 10. 2 3 54 3 11. 3 x 3 x 3 2 12. p 2 2 13. 5 M. Ruvalcaba Math 90 16.6 Applications with Right Triangles A. Right Triangles * A right triangle is a triangle with a 90 angle, as shown in the figure below. The small square in the corner indicates the 90 angle. Hypotenuse c a Leg b Leg * In a right triangle, the longest side is called the hypotenuse. It is also the side opposite the right angle. The other two sides are called legs. We generally use the letters a and b for the lengths of the legs and c for the length of the hypotenuse. They are related as follows. The Pythagorean Theorem: In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a 2 b 2 c 2 . Ex. Find the length of the unknown side of each right triangle. 1. 4 c 5 2. 10 b 12 3. 10 15 a M. Ruvalcaba Math 90 17.1 Introduction to Quadratic Equations A. Standard Form The following are quadratic equations. They contain polynomials of second degree. 4x2 7x 5 0 , 1 3t 2 t 9 , 2 5 y 2 6 y The quadratic equation 4 x 2 7 x 5 0 is said to be in standard form. Although the quadratic equation 4 x 2 5 7 x is equivalent to the preceding equation, it is not in standard form. Quadratic Equation: A quadratic equation is an equation equivalent to an equation of the type a x 2 b x c 0 , a 0 , where a, b, and c are real- number constants. We say that the preceding is the standard form of a quadratic equation. Ex. Write in standard form and determine a, b, and c. 1. 4x2 7x 5 0 a ____ , b ____ , c ____ 2. 4 y 2 5 y a ____ , b ____ , c ____ B. Solving Quadratic Equations of the Type ax 2 + bx = 0 * Sometimes we can use factoring and the principle of zero products to solve quadratic equations. Note: When c 0 and b 0 , we can always factor and use the principle of zero products. Ex. Solve. 3. 7x2 2x 0 M. Ruvalcaba Math 90 4. 4x 8x 0 2 C. Solving Quadratic Equations of the Type ax 2 + bx + c = 0 * When neither b nor c is 0, we can sometimes solve by factoring. Ex. Solve. 5. 2 x 2 x 21 0 6. y 3 y 2 6 y 3 * Recall that to solve a rational equation, we multiply both sides by the LCM of all the denominators in the equation. We may obtain a quadratic equation after a few steps. When that happens, we know how to finish solving, but we must still remember to check possible solutions because a replacement may result in division by 0. 3 5 7. 2 x 1 x 1 M. Ruvalcaba Math 90 17.2 Solving Quadratic Equations by Completing the Square A. Solving Quadratic Equations of the Type ax 2 = p * For equations of the type a x 2 p , we first solve for x 2 and then apply the principle of square roots, which states that a positive number has two square roots. The Principle of Square Roots: * The equation x 2 d has two real solutions when d 0 . The solutions are d and d . * The equation x 2 d has no real-number solution when d 0 . * The equation x 2 0 has 0 as its only solution. Ex. Solve. 1 2 1. x2 3 2. x 0 8 3. 3 x 2 7 0 4. 2x2 3 0 M. Ruvalcaba Math 90 Solving Quadratic Equations of the Type x + c = d 2 B. * In an equation of the type x c d , we have the square of a binomial equal to a 2 constant. We can use the principle of square roots to solve such an equation. Ex. Solve. x 5 9 x 2 7 2 2 5. 6. x 3 16 x 4 11 2 2 7. 8. * In Examples 5 through 8, the left sides of the equations are squares of binomials. If we can express an equation in such a form, we can proceed as we did in those examples. Ex. Solve. 9. x 2 8 x 16 49 10. x 2 6 x 9 64 M. Ruvalcaba Math 90 C. Completing the Square * We have seen that a quadratic equation like x 5 9 can be solved by using the 2 principle of square roots. We also noted that an equation like x 2 8 x 16 49 can be solved in the same manner because the expression on the left side (perfect square trinomial) is the square of a binomial, x 4 . This second procedure is the basis for 2 a method of solving quadratic equations called completing the square. * Suppose we have the following quadratic equation: x 2 10 x 4 . If we could add to both sides of the equation a constant that would make the expression on the left a perfect square trinomial, we could then solve the equation using the principle of square roots. * By adding 25 to the left side of the equation x 2 10 x , we can „force‟ a perfect trinomial square to be present. Adding 25 to the left side of the equation would mean that we would have to add the same amount to the right side as well. The equation would be solved as follows: x 2 10 x ______ 4______ x 2 10 x 25 4 25 x 5 29 2 x5 29 x 5 29 Completing the Square: To complete the square of an expression like x 2 b x , we take half of the coefficient of x and square it. Then we add that amount to both sides of the equation so that the left side becomes a perfect square trinomial. Ex. Solve. 11. x2 6x 8 0 12. x2 6x 8 0 M. Ruvalcaba Math 90 13. x 4x 7 0 2 14. x 12 x 23 0 2 15. x 2 3 x 10 0 16. x 2 3 x 10 0 17. 2 x 2 3x 1 18. 2 x 2 3x 3 0 M. Ruvalcaba Math 90 17.3 The Quadratic Formula A. Solving Using the Quadratic Formula The Quadratic Formula: The solutions of a x 2 b x c 0 are given by b b 2 4 ac x 2a Ex. Solve using the quadratic formula. 1. 5 x 2 8 x 3 2. x 2 3 x 10 0 3. x2 4x 7 4. x 2 x 1 M. Ruvalcaba Math 90 17.5 Applications and Problem Solving A. Using Quadratic Equations to Solve Applied Problems Ex. Solve. 1. The area of a rectangular red raspberry patch is 76 ft 2 . The length is 7 ft longer than three times the width. Find the dimensions of the raspberry patch. 2. The length of a rectangular area rug is 3 ft greater than the width. The area is 70 ft 2 . Find the length and the width. 3. A square is a carpenter‟s tool in the shape of a right triangle. One side, or leg, of a square is 8 in. longer than the other. The length of the hypotenuse is 8 13 in. Find the lengths of the legs of the square. M. Ruvalcaba Math 90 4. The current in a stream moves at a speed of 2 km / h . A boat travels 24 km upstream and 24 km downstream in a total time of 5 hr . What is the speed of the boat in still water? 5. The speed of a boat in still water is 10 km / h . The boat travels 12 km upstream and 28 km downstream in a total time of 4 hr . What is the speed of the stream? M. Ruvalcaba Math 90 17.6 Graphs of Quadratic Equations A. Graphing Quadratic Equations of the Type y = ax 2 + bx + c & B. Finding the x-Intercepts of a Quadratic Equation Ex. Graph: y x 2 Ex. Graph: y x 2 y 6 x y 5 x y 4 3 2 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 First find these 4 items of information… 1. Vertex 2. y-intercept 3. Whether the parabola opens up or down 4. x-intercept(s) b 1. To find the vertex, use the vertex formula ( x ) to find the x-coordinate. Then, substitute the 2a value of the x-coordinate back into the original quadratic equation to find the y-coordinate. 2. The y-intercept(where the parabola intersects the y-axis) is simply the “c”, i.e. the constant. 3. If the “a” is positive, the parabola opens up and if the “a” is negative, the parabola opens down. 4. To find the x-intercept(s), replace the “y” with a 0 and solve the remaining equation for “x”. Ex. Graph: y x 2 2 x 3 y 6 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 M. Ruvalcaba Math 90 Ex. Graph the quadratic equation. Label the ordered pairs for the vertex and the y-intercept. y 6 1. y x 1 2 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 y 2. y x2 2x 6 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 3. y x2 2x 1 6 y 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 4. y x2 2x 3 6 y 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 M. Ruvalcaba