Chapter 4: Rational Expressions and Equations

Transcription

1 Chapter 4: Rational Expressions and Equations Section 4.1 Chapter 4: Rational Expressions and Equations Section 4.1: Equivalent Rational Expressions Terminology: Rational Expression: An algebraic fraction with a numerator and a denominator that are polynomials. Examples include: 1 m y 2 1 x 2 2m + 1 y 2 x y + 1 Note that x 2 1 is a rational expression with a denominator of 1. Non-Permissible Values (NPV): The value of a variable that makes the denominator of a rational expression equal to zero. Example: In the expression x, the expression becomes undefined when x= -15, because this creates a zero denominator. Therefore, x= -15 is a non-permissible value for this expression. Determining Equivalent Rational Expressions (a) Write a rational number that is equivalent to 8 12 (b) Write a rational expression that is equivalent to 4x2 +8x 4x 93

2 Chapter 4: Rational Expressions and Equations Section 4.1 NOTE: When I create an equivalent rational expression, I must ensure to keep the nonpermissible value(s) for the expression equal to what we started with. Ie. Multiplying numerator and denominator by a factor with the same NPV or reducing the expression but still stating the original NPV. (c) Write a rational number that is equivalent to 6 18 (d) Write a rational expression that is equivalent to 3x3 +5x 4x Determining the Non-Permissible Values for a Rational Expression Ex: Determine the non-permissible value(s) for each rational expression, and then state all restrictions. (a) 4x3 4 x (b) 15 x 2 5x (c) 5x x+6 (d) x+7 x 2 +2x 94

3 Chapter 4: Rational Expressions and Equations Section 4.1 Determining if Rational Expressions are Equivalent To determine if two rational expressions are equivalent we must perform two checks: #1: We must check to see if both expressions have the same non-permissible values. If they do not, then they are not equivalent. If they are, move on to the second check. #2: Compare the rational expressions to ensure that one can be multiplied or divided by a number or variable to create the other. Ex: For each of the following, determine if the rational expressions are equivalent. (a) 9 3x 1 and x (b) 2 2x 4x and x 1 2x (c) x2 1 4x and 5 5x 3x Practice Problems #1,3,4,5,9,16 pg

4 Chapter 4: Rational Expressions and Equations Section 4.1 Section 4.1: Factoring Quadratic Equations (from Gr.11) Factoring A Quadratic Expression A quadratic equation can be solved in many cases by factoring. There are four major ways to factor a trinomial of the form y = ax 2 + bx + c. These were covered in math 1201 so we shall do a quick review. 1. Factoring Using Product and Sum: Product and Sum can only be used in situations where a=1. In such cases you must determine your factors by concluding what possible combination of two numbers can multiply to c and add to b EXAMPLES: Factor The following a. x 2 + 6x + 8 b. k 2 7k 30 c. j j 42 d. f 2 9f + 20 To solve a quadratic equation like those above, one need only factor as we just did, then set each factor to zero and solve for the given variable: a. x 2 + 6x + 8 = 0 b. k 2 + 8k + 7 = 0 c. j 2 + 3j 54 = 0 NOTE: We call this the zero product property in which if the product of two real numbers is zero, then one or both of the numbers must be zero. 96

5 Chapter 4: Rational Expressions and Equations Section Factoring Using Decomposition: Decomposition can be used in situations where a 1 and a GCF cannot be removed. In such cases you must determine your factors by following these steps: STEP1: Conclude what possible combination of two numbers can multiply to a c and add to b. STEP2: Decompose your middle term into those two numbers. STEP3: Group the first set and second set of terms. Pull out the GCF (Greatest Common Factor) of each group. STEP4: Then factor out the common bracketed term. EXAMPLES: Factor The following a. 5x 2 7x 6 b. 3k 2 13k 10 c. 8j j 5 d. 15f 2 7f 2 To solve a quadratic equation like those above, one need only factor as we just did, then set each factor to zero and solve for the given variable: a. 5x 2 7x 6 = 0 b. 4k 2 21k + 20 = 0 c. 6j j 14 = 0 97

6 Chapter 4: Rational Expressions and Equations Section Factoring Using GCF: In some cases where a 1, a GCF can be removed from the situation and allow it to be factored using Product and Sum or via Decomposition (with slightly more manageable numbers). EXAMPLES: Solve The following a. 5x 2 10x = 0 b. 3k 2 9k 12 = 0 c. 20x 2 50x 30 = 0 d. 10x x + 12 = 0 e. 63x 2 56x = 0 f. 1 2 m2 + 3m + 4 = 0 g. 1 5 p2 2p + 5 = 0 h. 0.25q q 2 = 0 98

7 Chapter 4: Rational Expressions and Equations Section Difference of Squares: Difference of squares can only be used in situations where b=0. In such cases both the a and c values will be perfect squares and there is a subtraction symbol between them. Your resulting factors will be the square root of each term with a different sign between them. EXAMPLES: Factor The following a. x 2 9 b. 9k 2 25 c. 144j d. 12f 2 75 To solve a quadratic equation like those above, one need only factor as we just did, then set each factor to zero and solve for the given variable: a. x 2 9 = 0 b. 144j = 0 c. 36j = 0 d. 45f 2 80 = 0 99

8 Chapter 4: Rational Expressions and Equations Section 4.2 Section 4.2: Simplifying Radical Expressions Simplifying a Rational Expression A rational expression is considered to be simplified when all possible common factors have been removed from the numerator and denominator. Ex. Simplify the following rational expressions (a) 24a 2 (b) 18x 4 18a 3 36x 7 (c) 15x 3 5x 15x 3 (d) 3y 9y 2 6y 3 100

9 Chapter 4: Rational Expressions and Equations Section 4.2 (e) 6m 2 8m (f) 3a 3 3a 2 3m 3 4m 2 12a+12 Practice Problems 1,2,3,4,5,7,8 pg

10 Chapter 4: Rational Expressions and Equations Section 4.3 Section 4.3: Mult. And Dividing Rational Expressions Multiplying Rational Expressions When we multiply two rational expressions, we must multiply the numerator of the first expression with the numerator of the second. We then multiply the denominator of the first expression with the denominator of the second. This may require the use of distributive property. After the numerator and denominator are worked out, we go through the procedure of simplification, to get the final answer. It is also advised to determine the non-permissible values for the expression before we multiply. Ex. Simplify the following product (a) 2x 2 12x 15x 5x x 6 (b) 12x 3 4x3 +8x 2 3x 2 +6x 5 102

11 Chapter 4: Rational Expressions and Equations Section 4.3 (c) 8b 3 +4b 2 6b b b+3 (d) 2a 3 18a 8a 24 6a 3 3a 3 +a 4 103

12 Chapter 4: Rational Expressions and Equations Section 4.3 Dividing Rational Expressions When we divide two rational expressions, we must multiply the first expression with the reciprocal of the second. This may require the use of distributive property. After the numerator and denominator are worked out, we go through the procedure of simplification, to get the final answer. It is also advised to determine the non-permissible values for the expression before we multiply. There is an extra set of NPVs that occur in division since both the zeros of the numerator and denominator of the second expression exist on the bottom of a faction at one point throughout the calculation. (a) x 5 5 3x 2 9x 6x 18 (b) 2w 6w2 6w 24w+4w 2 9w 3 +54w 2 104

13 Chapter 4: Rational Expressions and Equations Section 4.3 (c) x3 +x 2 16 x2 +x 20x 10 (d) 30x2 +15x x 3 2x3 +x 2 x 2 3x (e) 4x2 1 x+2 4x2 +2x 8x 2 32 Practice Problems 1,2,3,4,5,6,7,15 pg

14 Chapter 4: Rational Expressions and Equations Section 4.4 Section 4.4: Adding and Subtracting Rational Expressions Adding and Subtracting Rational Expressions Adding and subtracting rational expressions is done in the much the same way as addition and subtraction of fractions. We must first determine a common denominator for both expressions. Once we determine the LCD, we multiply both expressions by the required factors to ensure both have the same denominator. Once this is done we simply add or subtract the numerators, leaving the denominator alone. We must also state any NPVs for the expression. Ex1. Determine the sum (a) 3 8x x (b) 3 6x x 106

15 Chapter 4: Rational Expressions and Equations Section 4.4 Ex2. Determine the difference (a) 3n 2n+1 4 n 3 (b) 6 n 3 4 n+2 107

16 Chapter 4: Rational Expressions and Equations Section 4.4 Ex3. Simplify Sometimes it is necessary to use factoring techniques to help with simplification (a) x 2 16 x+4 (b) 2x x x 1 Practice Questions 3,4,5,6,7,8,9,11 pg

17 Chapter 4: Rational Expressions and Equations Section 4.5 Section 4.5: Solving Rational Equations Solving a Problem that Involves a Rational Equation Ex1. Salt water is flowing into a large tank that contains pure water. The concentration of salt in the tank, c, in grams per litre (g/l), at time t, in minutes, is given by the formula: c = 10t 25 + t Determine the time when the salt concentration in the tank reaches 3.75 g/l. Terminology: Rational Equation: An equation that involves one or more rational expressions. Ex. 5 x = 4 x

18 Chapter 4: Rational Expressions and Equations Section 4.5 Solving a Rational Equations Note: When solving a rational equation, we have to be on the lookout for extraneous roots. This means we must watch out for when the roots of the equation has an answer that is also a NPV, making it extraneous. Ex. Solve each rational equation. (a) 1 3x + 1 x = 1 6 (b) 1 x 1 x+1 =

19 Chapter 4: Rational Expressions and Equations Section 4.5 (c) z z 1 1 z 1 = 4 (d) 2 m m = 6 m 2 3m 111

20 Chapter 4: Rational Expressions and Equations Section 4.5 (e) 18 = 6 5 x 2 3x x 3 x (f) 2 a2 +4 = a+2 a 2 4 a 2 a 112

21 Chapter 4: Rational Expressions and Equations Section 4.5 Solving Rational Equations With Inadmissible Values Ex1: When they work together, Stuart and Lucy can deliver flyers to all the homes in the neighbourhood in 42 minutes. When Lucy works alone, she can deliver flyers in 13 minutes less than when Stuart works alone. When Stuart works alone, how long does he tale to deliver the flyers. Ex2. Two friends share a paper route. Sheena can deliver the papers in 40 minutes. Jeff can cover the same route in 50 minutes. How long to the nearest minute. Does the paper route take if they work together? 113

22 Chapter 4: Rational Expressions and Equations Section 4.5 Ex3. Stella takes 4 hours to paint a room. It takes Jose 3 h to paint the same area. How long will the paint job take if they work together? Using Rational Equations to Model and Solve a Problem Ex1. Rima bought a case of concert T-shirts for $450. She kept two T-shirts for herself and sold the rest for $560, making $10 on each T-shirt. How many were in the case? 114

23 Chapter 4: Rational Expressions and Equations Section 4.5 Ex. Jack also bought a case of concert T-shirts for $450. He kept two T-shirts for himself and sold the rest for $560, making $12 on each T-shirt. How many were in his case? Ex. Lydia frequently drives 189 km to visit friends in Canmore, Alberta. She noticed that she saves 36 minutes if she travels 24 km/h faster than her average speed. Determine her average speed. 115

24 Chapter 4: Rational Expressions and Equations Section 4.5 Ex. We estimate the cost of a grad trip to be $5400, which will be divided evenly among all the grads. At the last minute, 5 students decide not to go, so each of the remaining grads had to pay an additional $15. How many grads went on the trip? Practice Questions: 5,6,7,8,9,10,11,12,13,14,15,16,17,18 pg