6.3 The Binomial Theorem

Transcription

1 COMMON CORE L L R R L R Locker LESSON 6.3 The Binomial Theorem Name Class Date 6.3 The Binomial Theorem Common Core Math Standards The student is expected to: COMMON CORE A-APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. Also A-APR.A., S-CP.A., S-CP.B.7 Mathematical Practices COMMON CORE MP.7 Using Structure Language Objective Work with a partner to create a poster of terms used in this lesson, such as Binomial Theorem, binomial probability, binomial experiment and Pascal s Triangle. ENGAGE Essential Question: How is the Binomial Theorem useful? Possible answer: You can use the Binomial Theorem to expand a whole-number power of a binomial. The theorem gives the terms of the expansion as a product of a binomial coefficient (based on Pascal s Triangle), a power of one of the terms of the binomial, and a power of the other term in the binomial. The Binomial Theorem is also useful in calculating probabilities for a binomial experiment. PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo and what it means for a player to be an 85% free-throw shooter. Then preview the Lesson Performance Task. Essential Question: How is the Binomial Theorem useful? Explore Generating Pascal s Triangle Pascal s Triangle is a famous number pattern named after the French mathematician Blaise Pascal ( ). You can use Pascal s Triangle to help you expand a power of a binomial of the form (a + b ) n. Use the tree diagram shown to generate Pascal s Triangle. Notice that from each node in the diagram to the nodes immediately below it there are two paths, a left path (L) and a right path (R). You can describe a path from the single node in row 0 to any other node in the diagram using a string of Ls and Rs. First, notice that there is only one possible path to each node in row, which is why a appears in those nodes. In row 2, there is only one possible path, LL, to the first node and only one possible path, RR, to the last node, but there are two possible paths, LR and RL, to the center node. A B C Complete only rows 3 and 4 of Pascal s Triangle. (You will complete rows 5 and 6 in Step C.) In each node, write the number of possible paths from the top down to that node. Row 0 : Row : Row 2 : Row 3 : Row 4 : Row 5 : Row 6 : Look for patterns in the tree diagram. What is the value in the first and last node in each row? For every other node, the value in the node is the of the two values above it. Using the patterns in Step B, go back to Pascal s Triangle in Step A and complete rows 5 and 6. Answers for this step are shown in Step A. L R L R L R sum Resource Locker Module Lesson 3 Name Class Date 6.3 The Binomial Theorem Essential Question: How is the Binomial Theorem useful? A-APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y) n with coefficients determined by Pascal s Triangle. Also A-APR.A., S-CP.A., S-CP.B.7 Explore Generating Pascal s Triangle Complete only rows 3 and 4 of Pascal s Triangle. (You will complete rows 5 and 6 in Step C.) In each node, write the number of possible paths from the top down to that node. Resource Pascal s Triangle is a famous number pattern named after the French mathematician Blaise Pascal ( ). You can use Pascal s Triangle to help you expand a power of a binomial of the form (a + b ) n. Use the tree diagram shown to generate Pascal s Triangle. Notice that from each node in the diagram to the nodes immediately below it there are two paths, a left path (L) and a right path (R). You can describe a path from the single node in row 0 to any other node in the diagram using a string of Ls and Rs. First, notice that there is only one possible path to each node in row, which is why a appears in those nodes. In row 2, there is only one possible path, LL, to the first node and only one possible path, RR, to the last node, but there are two possible paths, LR and RL, to the center node. Row 0 : Row : Row 2 : 2 Row 3 : 3 3 Row 4 : Row 5 : Row 6 : HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. Look for patterns in the tree diagram. What is the value in the first and last node in each row? sum For every other node, the value in the node is the of the two values above it. Using the patterns in Step B, go back to Pascal s Triangle in Step A and complete rows 5 and 6. Answers for this step are shown in Step A. Module Lesson Lesson 6.3

2 Reflect. Using strings of Ls and Rs, write the paths that lead to the second node in row 3 of Pascal s Triangle. How are the paths alike, and how are they different? The paths that lead to the second node in row 3 are LLR, LRL, and RLL. These paths contain the same elements, but in different orders. 2. The path LLRLR leads to which node in which row of Pascal s Triangle? What is the value of that node? The path LLRLR leads to the third node in row 5. This node has a value of 0. Explore 2 Relating Pascal s Triangle to Powers of Binomials As shown, the value in position r of row n of Pascal s Triangle is written as n C r, where the position numbers in each row start with 0. In this Explore, you will see how the values in Pascal s Triangle are related to powers of a binomial. EXPLORE Generating Pascal s Triangle INTEGRATE TECHNOLOGY Students have the option of completing the Explore activity either in the book or online. A Row 0 : 0C 0 Row : C 0 C Row 2 : 2C 0 2 C 2 C 2 Row 3 : 3C 0 3 C 3 C 2 3 C 3 Expand each power. (a + b) 0 = (a + b) = (a + b) 2 = a + b a 2 + 2ab + b 2 Square of a binomial (a + b) 3 = a 3 + 3a 2 b+ 3ab 2 + b 3 Multiply (a + b) 2 by (a + b). QUESTIONING STRATEGIES Which row of Pascal s Triangle has 2 values? row What do you notice if you read any row of Pascal s Triangle from right to left? The values are the same as when read from left to right. Each row is symmetric about a vertical line through the number at the top of the triangle. (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 Multiply (a + b) 3 by (a + b). B Identify the patterns in the expanded form of (a + b) n. The exponents of a start at n and [increase/decrease] by each term. The exponents of b start at 0 and [increase/decrease] by each term. The sum of the exponents in each term is n. The coefficients of the terms in the expanded form of (a + b) n are the values in row n of Pascal s Triangle. Reflect 3. How many terms are in the expanded form of (a + b) n? There are n + terms in the expanded form of (a + b) n. 4. Without expanding the power, determine the middle term of (a + b) 6. Explain how you found your answer. Since the coefficient is 6 C 3 = 20, the exponents of a and b have a sum of 6, and the exponents of the middle term are the same when n is even, the middle term must be 20 a 3 b Without expanding the power, determine the first term of (a + b) 5. Explain how you found your answer. The first term of (a + b) 5 is a 5 since the first term of (a + b) n is always a n. Module Lesson 3 PROFESSIONAL DEVELOPMENT Learning Progressions In the previous lesson, students learned how to multiply polynomials. They also learned the rules for finding squares and cubes of binomials. In this lesson, students continue their work with multiplication of polynomials, examining the patterns formed when binomials are raised to even higher powers. They explore the patterns found in Pascal s Triangle, and learn how to use the Binomial Theorem to expand powers of binomials. Students also apply the Binomial Theorem to a situation involving binomial probabilities. EXPLORE 2 Relating Pascal s Triangle to Powers of Binomials QUESTIONING STRATEGIES What position in Pascal s Triangle is represented as 5 C 2? Explain. Position 2 of row 5; n C r represents position r of row n. What number in row 5 of Pascal s Triangle is the same number as 5 C 4? Explain how you know. 5 C ; Because the pattern is symmetric, the number in the fourth position in the row will be the same as that in the fourth position from the end of the row. The Binomial Theorem 298

3 EXPLAIN Expanding Powers of Binomials Using the Binomial Theorem QUESTIONING STRATEGIES What must be true about the degree of each term in the expansion? It must be equal to the exponent to which the binomial is raised. Will the coefficient of the second term of every binomial raised to the fourth power always be 4? Explain. No. If the coefficient of one or both of the terms in the binomial is not, the coefficient in the expansion will be 4 times the coefficient of the first term raised to the third power times the coefficient of the second term. Explain Expanding Powers of Binomials Using the Binomial Theorem The Binomial Theorem states the connection between the terms of the expanded form of (a + b) n and Pascal s Triangle. Binomial Theorem For any whole number n, the binomial expansion of (a + b) n is given by (a + b) n = nc 0a n b 0 + nc a n - b + nc 2a n - 2 b nc n - a b n - + nc na 0b n where nc r is the value in position r (where r starts at 0) of the nth row of Pascal s Triangle. Since it can be cumbersome to look up numbers from Pascal s Triangle each time you want to expand a power of a binomial, you can use a calculator instead. To do so, enter the value of n, press MATH, go to the PRB menu, select 3:nCr, and then enter the value of r. The calculator screen shows the values for 6 C, 6C 2, and 6C 3. Example (x - 2)3 Use the Binomial Theorem to expand each power of a binomial. Step Identify the values in row 3 of Pascal s Triangle., 3, 3, and Step 2 Expand the power as described by the Binomial Theorem, using the values from Pascal s Triangle as coefficients. (x - 2) 3 = x 3 (-2) 0 + 3x 2 (-2) + 3x (-2) 2 + x 0 (-2) 3 Step 3 Simplify. AVOID COMMON ERRORS When expanding a power of a binomial like (2x + 3) 4, students may raise the x in 2x to appropriate powers but not the coefficient 2. For instance, they may think that the first term in the expansion of (2x + 3) 4 is 2 x 4 instead of (2x) 4 = 6 x 4. Similarly, students may forget to use the numbers from Pascal s Triangle in an expansion. For example, they may think that the second term in the expansion of (2x + 3) 4 is (2x) 3 (3) = 24 x 3 instead of 4 (2x) 3 (3) = 96 x 3. (x + y) 7 (x - 2) 3 = x 3-6x 2 + 2x - 8 Step Use a calculator to determine the values of 7 C 0, 7C, 7C 2, 7C 3, 7C 4, 7C 5, 7C 6, and 7C 7. Step 2 Expand the power as described by the Binomial Theorem, using the values of 7 C 0, 7C, 7C 2, 7C 3, 7C 4, 7C 5, 7C 6, and 7C 7 as coefficients. (x + y) 7 = x y + 7 x y + 2 x y + 35 x y Step 3 Simplify x y + 2 x y + 7 x y + x y (x + y) 7 = x + 7 x y + 2 x y + 35 x y + 35 x y x y + 7 xy + y, 7, 2, 35, 35, 2, 7, and INTEGRATE TECHNOLOGY If the binomial contains only one variable, students can use a graphing calculator to check the accuracy of the expansion. Students can enter the original expression and the expanded form as two separate functions and check to see that the graphs are identical. Module Lesson 3 COLLABORATIVE LEARNING Small Group Activity Have students work in groups of 3 or 4. Instruct each group to design an experiment that models the situation in Example 2. Have them perform their experiment 50 times, recording the result of each trial. Have them use their results to verify the probabilities calculated using the Binomial Theorem. Have each group present their model and their results to the class. 299 Lesson 6.3

4 Reflect 6. What happens to the signs of the terms in the expanded form of (x - 2) 3? Why does this happen? The signs alternate between positive and negative because -2 to an even power is positive while -2 to an odd power is negative. 7. If the number is written as the binomial (0 + ), how can you use the Binomial Theorem to find 2, 3, and 4? What is the pattern in the digits? 2 = (0)() + 2 = 2 3 = (0) 2 () + 4 (0)() = 33 4 = (0) 3 () + 6 (0) 2 () (0)() = 4,64 The digits are the values in the rows of Pascal s Triangle. Your Turn 8. Use the Binomial Theorem to expand (x - y) 4. The values in row 4 of Pascal s Triangle are, 4, 6, 4, and. (x - y) 4 = x 4 (-y) 0 + 4x 3 (-y) + 6 x 2 (-y) x (-y) 3 + x 0 (-y) 4 = x 4-4x 3 y + 6x 2 y 2-4xy 3 + y 4 EXPLAIN 2 Solving a Real-World Problem Using Binomial Probabilities CONNECT VOCABULARY Have students relate the prefix bi- to the number two. A binomial expression has two terms. A binomial experiment has two outcomes. This connection should help students understand some of the essential terms in this lesson. Relate the prefix to binomial probability and the Binomial Theorem as well. Explain 2 Solving a Real-World Problem Using Binomial Probabilities Recall that the probability of an event A is written as P (A) and is expressed as a number between 0 and, where 0 represents impossibility and represents certainty. When dealing with probabilities, you will find these two rules helpful. QUESTIONING STRATEGIES Why are the probabilities multiplied by nc r? because that is the number of different ways the outcome could occur. Addition Rule for Mutually Exclusive Events: If events A and B are mutually exclusive (that is, they cannot occur together), then P (A or B) = P (A) + P (B). For example, when rolling a die, getting a and getting a 2 are mutually exclusive events, so P ( or 2) = P () + P (2) = = Complement Rule: The complement of event A consists of all of the possible outcomes that are not part of A, and the probability that A does not occur is P (not A) = - P (A). For example, when rolling a die, the probability of not getting a 2 is P (not 2) = - P (2) = - 6 = 5 6. A binomial experiment involves many trials where each trial has only two possible outcomes: success or failure. If the probability of success in each trial is p and the probability of failure in each trial is q = - p, the binomial probability of exactly r successes in n trials is given by P (r) = n C r p r q n - r. Since n C r = n C n - r, you can rewrite P (r) as P (r) = n C n - r p r q n - r, which represents the (n - r) th term in the expanded form of (p + q) n. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP. Students have experience calculating probabilities for various types of experiments. Reinforce that in binomial experiments, each trial has only two possible outcomes. Compare this to the rolling of a number cube for which there are six possible outcomes. However, if the experiment is such that only rolling a 6 is considered a success, then the experiment is a binomial experiment. Module Lesson 3 DIFFERENTIATE INSTRUCTION Kinesthetic Experience Kinesthetic learners may benefit from moving a physical object, such as a coin or other marker, along the paths in Pascal s Triangle. For instance, you could give each of 6 students a unique pathway to follow using a four-letter sequence of Ls and Rs, such as RRLR. Those 6 students would then move their markers along the paths to determine at what node in row 4 of the triangle they land. By surveying the 6 students, you would find how many students landed at each of the five nodes in row 4, thereby generating the numbers in that row of Pascal s Triangle. AVOID COMMON ERRORS Students may sometimes confuse p and q. Remind them that p is the probability of success and that q is the probability of failure, p. Have them note that the sum of the probabilities they enter in the formula must be. Prompt them to consider and discuss why this is so. For visually challenged students, have them consider that the loop in p faces forward, the loop in q faces backward. The Binomial Theorem 300

5 Example 2 One in 5 boats traveling down a river bypass a harbor at the mouth of the river and head out to sea. Currently, 4 boats are traveling down the river and approaching the mouth of the river. What is the probability that exactly 2 of the 4 boats head out to sea? The probability that a boat will head out to sea is _ 5, or 0.2. Substitute 4 for n, 2 for r, 0.2 for p, and 0.8 for q. P (2) = 4 C 2 (0.2) (0.8) = 6 (0.2) 2 (0.8) 2 = 6 (0.04) (0.64) = So, the probability that exactly 2 of the 4 boats will head out to sea is 0.536, or 5.36%. What is the probability that at least 2 of the 4 boats will head out to sea? To find the probability that at least 2 of the 4 boats will head out to sea, find the probability that 2, 3, or 4 boats will head out to sea and add the probabilities. From Part A, you know that P (2) = Image Credits: Johnny Lye/Shutterstock 3 P (3) = 4 C (0.2) (0.8) 3 = 4 ( )( 0.8 ) = P (4) = 4 C (0.2) (0.8) 4 = ( )( ) = P (at least 2) = P (2 or 3 or 4) = P (2) + P (3) + P (4) = = So, the probability that at least 2 of the 4 boats will head out to sea is 0.808, or 8.08%. Module 6 30 Lesson 3 LANGUAGE SUPPORT Communicate Math Have students work in pairs to create and complete a poster with a large table showing the information below. Term Binomial Theorem Binomial probability Binomial experiment Pascal s Triangle Picture or Equation Explanation 30 Lesson 6.3

6 Reflect 9. In words, state the complement of the event that at least 2 of the 4 boats will head out to sea. Then find the probability of the complement. The complement of the event that at least 2 of the 4 boats will head out to sea is the event that fewer than 2 boats will head out to sea. To find the probability of the complement, subtract the probability found in Part B from. P (fewer than 2) = = So, the probability that fewer than 2 boats will head out to sea is 0.892, or 8.92%. Your Turn 0. Students are assigned randomly to of 3 guidance counselors at a school. What is the probability that Ms. Banks, one of the school s guidance counselors, will get exactly 2 of the next 3 students assigned? The probability that a student will be assigned to Ms. Banks is _ 3. P (2) = 3 C 2 ( _ = 3 ( _ Elaborate 3) 2 ( 2_ 3) 9 )( 2_ 3) = 2_ The probability that Ms. Banks will get exactly 2 of the next 3 students is 2_ 9, or about 22.2%.. How do the numbers in one row of Pascal s Triangle relate to the numbers in the previous row? The first and last numbers in a row of Pascal s Triangle are both. Every other number is the sum of the two numbers immediately above it in the previous row. 2. How does Pascal s Triangle relate to the power of a binomial? For a binomial raised to the nth power, each term in the expanded form of the power includes a number in row n of Pascal s Triangle as a factor. 3. The expanded form of (p + q) 3 is p 3 + 3p 2 q + 3pq 2 + q 3. In terms of a binomial experiment with a probability p of success and a probability q of failure on each trial, what do each of the terms p 3, 3p 2 q, 3p q 2, and q 3 represent? The terms p 3, 3p 2 q, 3 pq 2, and q 3 represent the probabilities of exactly 3, 2,, and 0 successes, respectively, in 3 trials of a binomial experiment. 4. Essential Question Check-In The Binomial Theorem says that the expanded form of (a + b) n is a sum of terms of the form n C r a n - r b r for what values of r? The values of r are 0,, 2,, n. ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Have students discuss why the number n C r for particular values of n and r in Pascal s Triangle is the same number used for the coefficient of a binomial probability of exactly r successes in n trials. Prompt them to consider how they found the numbers in the triangle initially in the Explore, and the relationship to the meaning of the coefficient in the binomial probability. CONNECT VOCABULARY Ask students to use their own words to explain what mutually exclusive events are and to provide an example. SUMMARIZE THE LESSON How do you use the Binomial Theorem to expand powers of binomials? To expand a binomial raised to the nth power, identify the values in row n of Pascal s Triangle. Expand the power as described in the Binomial Theorem, using the values from Pascal s Triangle as multipliers of the terms. Then simplify each term. Module Lesson 3 The Binomial Theorem 302

7 EVALUATE Evaluate: Homework and Practice. The path LLRRLLR leads to which node in which row of Pascal s Triangle? What is the value of that node? Online Homework Hints and Help Extra Practice The path LLRRLLR leads to the fourth node in row 7. This node has a value of = 35 when Pascal s Triangle in Explore is extended to row 7. ASSIGNMENT GUIDE Concepts and Skills Explore Generating Pascal s Triangle Explore 2 Relating Pascal s Triangle to Powers of Binomials Example Expanding Powers of Binomials Using the Binomial Theorem Example 2 Solving a Real-World Problem Using Binomial Probabilities Practice Exercises, 24 Exercise 25 Exercises 2 8, 23 Exercises 9 22, Without expanding the power, determine the middle term of (a + b) 8. Explain how you found your answer. Since the coefficient is 8 C 4 = 70, the exponents of a and b have a sum of 8, and the exponents of the middle term are the same when n is even, the middle term must be 70 a 4 b 4. Use the Binomial Theorem to expand each power of a binomial. 3. (x + 6) 3 (x + 6) 3 = x 3 (6) 0 + 3x 2 (6) + 3 x (6) 2 + x 0 (6) 3 4. (x - 5) 4 5. (x + 3) 6 = x 3 + 8x x + 26 (x - 5) 4 = x 4 (-5) x 3 (-5) + 6 x 2 (-5) x (-5) 3 + x 0 (-5) 4 = x 4-20x x 2-500x (x + 3) 6 = x 6 (3) x 5 (3) + 5x 4 (3) x 3 (3) x 2 (3) 4 + 6x (3) 5 + x 0 (3) 6 = x 6 + 8x x x x x COOPERATIVE LEARNING Have students work in pairs to practice constructing Pascal s Triangle. Encourage them to look for patterns in the resulting triangle that can help them readily obtain the coefficients they need for any given binomial expansion or probability. 6. (2x - ) 3 (2x - ) 3 = (2x) 3 (-) (2x) 2 (-) + 3 (2x) (-) 2 + (2x) 0 (-) 3 = 8 x 3-2 x 2 + 6x - 7. (3x + 4) 5 (3x + 4) 5 = (3x) 5 (4) (3x) 4 (4) + 0 (3x) 3 (4) (3x) 2 (4) (3x) (4) 4 + (3x) 0 (4) 5 = 243 x x x x x (2x - 3) 7 (2x - 3) 7 = (2x) 7 (-3) (2x) 6 (-3) + 2 (2x) 5 (-3) (2x) 4 (-3) (2x) 3 (-3) (2x) 2 (-3) (2x) (-3) 6 + (2x) 0 (-3) 7 = 28x x x 5-5,20x ,680x 3-20,42x 2 + 0,206x -287 Module Lesson 3 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 2 Recall of Information MP.7 Using Structure 3 6 Recall of Information MP.5 Using Tools 7 8 Recall of Information MP.2 Reasoning Skills/Concepts MP.2 Reasoning 2 Recall of Information MP.7 Using Structure Recall of Information MP.5 Using Tools 303 Lesson 6.3

8 9. (x + 2y) 5 (x + 2y) 5 = x 5 (2y) 0 + 5x 4 (2y) + 0x 3 (2y) x 2 (2y) 3 + 5x (2y) 4 + x 0 5 (2y) 0. (3x - y) 4. (5x + y) 4 2. (x - 6y) 5 3. (5x - 4y) 3 4. (4x + 3y) 6 = x 5 + 0x 4 y + 4 0x 3 y x 2 y xy y 5 (3x - y) = (3x) (-y) + 4 (3x) 3 (-y) + 6 (3x) 2 (-y) (3x) (-y) (3x) (-y) = 8 x 4-0 8x 3 y + 5 4x 2 y 2-2 xy 3 + y 4 (5x + y) 4 = (5x) 4 y (5x) 3 y + 6 (5x) 2 y (5x) y 3 + (5x) 0 y 4 = 625 x x 3 y + 50 x 2 y xy 3 + y 4 (x - 6y) 5 = x 5 (-6y) x 4 (-6y) + 0x 3 (-6y) 2 + 0x 2 (-6y) 3 + 5x (-6y) 4 + x 0 (-6y) 5 = x 5-30 x 4 y x 3 y x 2 y xy y 5 (5x - 4y) 3 = (5x) 3 (-4y) (5x) 2 (-4y) + 3 (5x) (-4y) 2 + (5x) 0 (-4y) 3 = 2 5x x 2 y xy 2-64y 3 (4x + 3y) 6 = (4x) 6 (3y) (4x) 5 (3y) +5 (4x) 4 (3y) (4x) 3 (3y) (4x) (3y) 4 +6 (4x) (3y) 5 + (4x) 0 (3y) 6 QUESTIONING STRATEGIES If a term in a binomial being raised to a power has a coefficient other than, what happens to that coefficient when the power is expanded using the Binomial Theorem? The coefficient is raised to the same power as the variable for each term and then multiplied by the corresponding value from Pascal s Triangle. AVOID COMMON ERRORS Students may use the coefficients from the wrong row of Pascal s Triangle when expanding powers of binomials. Remind them that the coefficients needed for the expansion of (a + b) n will be found in the row of Pascal s Triangle that contains n + numbers. = 4096 x 6 +8,432 x 5 y + 34, 560x 4 y ,56 0x 3 y 3 +9,44 0x 2 y xy y 6 Use the Binomial Theorem to find the specified term of the given power of a binomial. (Remember that r starts at 0 in the Binomial Theorem, so finding, say, the second term means that r =.) 5. Find the fourth term in the expanded form of (x - ) 6. Let n = 6, r = 3, a = x, and b = - in nc ra n-r b r. 6 C 3 x 6-3 (-) 3 = 20 x 3 (-) 3 = -20 x 3 6. Find the second term in the expanded form of (2x + ) 4. Let n = 4, r =, a = 2x, and b = in nc ra n-r b r. 4 C (2x) 4- () = 4 (2x) 3 () = 32 x 3 7. Find the third term in the expanded form of (3x - 2y) 5. Let n = 5, r = 2, a = 3x, and b = -2y in nc ra n-r b r. 5 C 2 (3x) 5-2 (-2y) 2 = 0 (3x) 3 (-2y) 2 = 080 x 3 y 2 8. Find the fifth term in the expanded form of (6x + 8y) 7. Let n = 7, r = 4, a = 6x, and b = 8y in nc ra n - r b r. 7 C 4 (6x) 7-4 (8y) 4 3 = 35 (6x) (8y) 4 = 30,965,760x 3 y 4 Module Lesson 3 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 24, 26 2 Skills/Concepts MP.3 Logic 25 3 Strategic Thinking MP.3 Logic The Binomial Theorem 304

9 VISUAL CUES Before they begin the expansion of a binomial raised to a power, some students may find it helpful to write out a fill-in-the-blank framework, such as ( + ) 4 = (_ ) 4 + 4(_ ) 3 (_) + 6(_ ) 2 (_ ) 2 + 4(_)(_ ) 3 + (_ ) 4. First, have them find the appropriate numbers from Pascal s Triangle. Then, they can fill in the exponents for the powers of the first term of the binomial in descending order: 4, 3, 2,, 0, and the exponents for the powers of the second term of the binomial in ascending order: 0,, 2, 3, 4. If the binomial has the form (a - b) n, remind students to alternate the signs of the terms in the expansion, starting with a positive first term. They can then enter the values of a and b and simplify, being sure to apply each exponent to both the variable and its coefficient for variable terms. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 To ensure that students understand the parameters that constitute a binomial experiment, have them brainstorm real-world situations that lend themselves to binomial probability contexts. Then ask them to discuss events for which it might be easier to find the probability of the complement of the event in order to determine the probability of the event itself, than to determine the probability of the event directly. Image Credits: Nataliya Hora/Shutterstock Ellen takes a multiple-choice quiz that has 5 questions, with 4 answer choices for each question. 9. What is the probability that she will get exactly 2 answers correct by guessing? The probability that Ellen will get any one answer correct is _ 4. P(2) = 5 C 2 ( _ 4) 2 ( 3_ 4) 3 = 0 ( 6) ( 27 64) = So, the probability that Ellen will get exactly 2 answers correct by guessing is 35, or about 26.4% What is the probability that Ellen will get at least 3 answers correct by guessing? P (3) = 5 C 3 ( _ 4) 3 ( 3_ 4) 2 = 0 ( P (4) = 5 C 4 ( _ 4) 4 ( 3_ 4) = 5 ( P(5) = 5 C 5 ( _ 4) 5 ( 3_ 4) 0 = ( P (at least 3) = P (3 or 4 or 5) = P (3) + P (4) + P (5) Manufacturing A machine that makes a part used in cars has a 98% probability of producing the part within acceptable tolerance levels. The machine makes 25 parts per hour. 2. What is the probability that the machine will make exactly 20 acceptable parts in an hour? 22. What is the probability that the machine makes 23 or fewer acceptable parts? 64) ( 6) 9 = ) ( 3_ 4) = ) () = 024 = = So, the probability that Ellen will get at least 3 answers correct by guessing is 53 or about 0.4%. 52 The probability that the machine will make an acceptable part is P (20) = 25 C 20 (0.98) 20 (0.02) The probability that the machine will produce exactly 20 acceptable parts is about , or about 0.04%. P (23 or fewer) = - P (24 or 25) = - [P (24) + P (25) ] = - [ 25 C 24 (0.98) 24 (0.02) + 25 C 25 (0.98) 25 (0.02) 0 ] - [ ] = So, the probability that the machine will produce 23 or fewer acceptable parts is about , or about 8.86%. Module Lesson Lesson 6.3

10 23. Match each term of an expanded power of a binomial on the right with the corresponding description of the term on the left. (Remember that r starts at 0 in the Binomial Theorem, so finding, say, the second term means that r =.) A. Fifth term in the expanded form of (x + 2) 6 B 640 x 2 B. Fourth term in the expanded form of (x + 4) 5 D 48 x 2 C. Third term in the expanded form of (x + 8) 4 A 240 x 2 D. Second term in the expanded form of (x + 6) 3 C 384 x 2 A. Fifth term: 6 C 4 x 6-4 (2) 4 = 5 x 2 (6) = 2 40x 2 B. Fourth term: 5 C 3 x 5-3 (4) 3 = 0x 2 (64) = 64 0x 2 C. Third term: 4 C 2 x 4-2 (8) 2 = 6x 2 (64) = 38 4x 2 D. Second term: 3 C x 3- (6) = 3 x 2 (6) = 48 x 2 H.O.T. Focus on Higher Order Thinking WORLD HISTORY Students may be interested to learn more about Blaise Pascal and Pascal s Triangle. Born in 623, Pascal was a child prodigy who was building calculating machines by the time he was a teenager. The computer programming language Pascal is named for him. Encourage students to look for other patterns in Pascal s Triangle, such as horizontal sums (the rows add to powers of 2) and patterns that form the Fibonacci sequence. 24. Construct Arguments Identify the symmetry in the rows of Pascal s Triangle and give an argument based on strings of Ls and Rs to explain why the symmetry exists. When presented in a tree diagram as in Explore, Pascal s Triangle has reflection symmetry in the vertical line that passes through the node at the top of the tree. This means that in any row n, nc r = nc n-r. The symmetry exists because for each string of Ls and Rs that takes you to a node on the left side of the line of symmetry, you can replace each L with an R and each R with an L to obtain a string that takes you to the corresponding node (in the same row) on the right side of the line of symmetry. For instance, in row 4, the string LLRL takes you to the node in position, while the mirror-image string RRLR takes you to the node in position 3. This means that there will be the same number of pathways to get to position as there are to get to position 3, so 4 C = 4 C Communicate Mathematical Ideas Explain why the numbers from Pascal s Triangle show up in the Binomial Theorem. If you think of a tree diagram like the one for Pascal s Triangle but with a and b substituted for L and R, you can see that a pathway from the top of the tree down to the node in, say, position 2 (counting from 0) of row 5, generates a string such as aabab, which is one way to produce the term with the variable part a 3 b 2 in the expanded form of (a + b) 5. (In essence, the string aabab tells you to use the a-terms from the first, second, and fourth factors in the product (a + b) (a + b) (a + b) (a + b) (a + b) and to use the b-terms from the third and fifth factors.) There are 9 other pathways that lead to position 2 of row 5: bbaaa, abbaa, aabba, aaabb, baaab, abaab, baaba, ababa, and babaa. So, there is a total of 0, or 5 C 2, ways to produce the term with the variable part a 3 b 2, which means that a 3 b 2 has a coefficient of 5 C 2. In general, the term in the expanded form of (a + b) n with the variable part a n-r b r has the coefficient nc r, which is what the Binomial Theorem tells you. Module Lesson 3 The Binomial Theorem 306

11 MULTIPLE REPRESENTATIONS In higher mathematics, the notation ( n r) is used instead of n C r. Students might create a version of Pascal s Triangle using this notation. 26. Represent Real-World Situations A small airline overbooks flights on the assumption that some passengers will not show up. The probability that a passenger shows up is 0.8. What number of tickets can the airline sell for a 20-seat flight and still have a probability of seating everyone that is at least 90%? Explain your reasoning. JOURNAL Have students describe the steps they would take to expand (3c + 2d) 4 using the Binomial Theorem. Image Credits: Ashley Cooper/Alamy For the probability that everyone is seated to be at least 90%, the probability that too many passengers show up must be no more than 0%. First, consider the case where 2 tickets are sold and 2 passengers show up: P (2 passengers show up) = 2 C 2 (0.8) 2 (0.2) 0 (0.0092)() = = 0.92% Next, consider the case where 22 tickets are sold and either 2 or 22 passengers show up: P (2 or 22 passengers show up) = P (2) + P (22) = 22 C 2 (0.8) 2 (0.2) + 22 C 22 (0.8) 22 (0.2) 0 22 (0.0092) (0.2) + (0.0074) () = = 4.79% Next, consider the case where 23 tickets are sold and 2, 22, or 23 passengers show up: P (2, 22, or 23 passengers show up) = P (2) + P (22) + P (23) = 23 C 2 (0.8) 2 (0.2) C 22 (0.8) 22 (0.2) + 23 C 23 (0.8) 23 (0.2) = = 3.3% So, the airline can sell 22 tickets for a 20-seat flight and still have a probability of seating everyone that is at least 90%. Module Lesson Lesson 6.3

12 Lesson Performance Task Suppose that a basketball player has just been fouled while attempting a 3-point shot and is awarded three free throws. Given that the player is 85% successful at making free throws, calculate the probability that the player successfully makes zero, one, two, or all three of the free throws. Which situation is most likely to occur? The number of trials, or free throws, the player gets is 3, so let n = 3. The player s probability of success is 0.85, so p = This means the player s probability of failure is = 0.5. So, q = 0.5. Find the binomial probabilities of making the possible numbers of free throws. Player makes zero free throws: The number of successes is 0, so r = 0. P(0) = 3 C 0 (0.85) 0 (0.5) 3-0 = () ( ) = The probability that the player makes zero free throws is %. Player makes one free throw: The number of successes is, so r =. P() = 3 C (0.85) (0.5) 3- = 3 (0.85) (0.0225) = The probability that the player makes one free throw is %. Player makes two free throws: The number of successes is 2, so r = 2. P(2) = 3 C 2 (0.85) 2 (0.5) 3-2 = 3(0.7225)(0.5) = The probability that the player makes two free throws is %. Player makes three free throws: The number of successes is 3, so r = 3. P(3) = 3 C 3 (0.85) 3 (0.5) 3-3 = (0.6425) () = The probability that the player makes all three free throws is 6.425%. Because 6.425% is the highest probability, the player is most likely to make all three free throws. AVOID COMMON ERRORS Students may neglect to include the binomial coefficients in their calculations. This will not cause incorrect probabilities when r = n or when r = 0, because nc r = in these cases, but it will cause incorrect probabilities in all other cases. Remind students that there is more than one way to make or miss 2 of 3 free throws, so the binomial coefficient must be used as a multiplier to account for these ways. It will help them remember to always write the binomial coefficient, even when they know the value is. INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Graphing calculators can evaluate binomial coefficients. To evaluate 7 C 3 using the appropriate keys and menus, use the sequence 7, ncr, 3 ENTER. Module Lesson 3 EXTENSION ACTIVITY Have students consider what free-throw shooting percentage gives a player better than a 50% chance of making all or none of any number of free throws. For example, have them find what free-throw shooting percentage gives a player a better than 50% chance of making both of two free throws, and what percentage gives a player better than a 50% chance of making all three of three free throws. Then have students consider a better than 50% chance of missing both or of missing all three. Students can generalize these results. When P > 0.5, the chance of making 2 of 2 free throws is greater than 50%, so P > 70.7%. When 3 P > 0.5, the chance of making 3 of 3 free throws is greater than 50%, so P > 79.4%. The chances of missing both or of missing all three are these numbers subtracted from 00%: 29.3% and 20.6%. Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. The Binomial Theorem 308