Objectives

This lesson will build on the topic of polynomials and their roots. Students will: - show how roots and factors are related. - determine whether a given value is a root of a polynomial using synthetic division.

Core Questions

- How are relationships expressed mathematically? - How are expressions, equations, and inequalities utilized to quantify, solve, model, and/or analyze mathematical problems? - How can mathematics help us communicate more effectively? - How may patterns be used to describe mathematical relationships? - What does it mean to evaluate or estimate a numerical quantity? - How can mathematics help to measure, compare, depict, and model numbers? - What factors determine whether a tool or method is appropriate for a specific task? - How can we know whether a real-world scenario should be represented as a quadratic, polynomial, or exponential function? - How would you explain the advantages of using multiple approaches to portray polynomial functions (tables, graphs, equations, and contextual situations)?

Vocabulary

- Factor: One of two or more expressions that are multiplied.  - Fundamental theorem of algebra: Every polynomial equation of degree n ≥ 1, with complex coefficients, has at least one root which is a complex number (real or imaginary).  - Root: The solution to a given equation.  - Synthetic division: A short way of dividing a polynomial by a binomial.  - Remainder: In division, the difference of the dividend minus the product of the quotient times the divisor.  - Zero of a function: The value of the argument for which the function is equal to zero; also the Root of the function and the Solution to the equation.

Materials

- poster paper - markers - copies of Lesson 2 Exit Ticket (M-A2-3-2_Lesson 2 Exit Ticket and KEY)

Assignment

- The Extension activity outcomes will show how well students can generalize synthetic division principles. Using literal terminology can cause students to become distracted by the requirement for disciplined notation. For students who struggle with the activity, consider substituting trial values with small integers to generate binomial outcomes.  - The Lesson 2 Exit Ticket activity (M-A2-3-2_Lesson 2 Exit Ticket and KEY) asks students to match the polynomial's unique roots to the expressions.

Supports

Active Engagement, Modeling, Explicit Instruction   W: This lesson teaches students about synthetic division and approaches for determining if a term is a factor of a polynomial equation using active engagement, modeling, and explicit instruction. The lesson also teaches students how to create their own representations of the link between polynomial roots and factors.  H: Calculating the quotient with a multi-digit divisor provides an ideal introduction to synthetic division techniques. The two procedures are similar in structure and consequence, and long division with a familiar algorithm provides students with confidence to try the new method.  E: Synthetic division can be performed with assurance by following the predefined processes of dividing, multiplying, subtracting, checking, and bringing down  R: In Activity 3 (pair activity), students design their own polynomials starting with the integers they chose as roots. Recognizing the roots in factored form requires thinking backward from previously practiced processes. Students must also evaluate another student's work using their own grasp of the technique.  E: The Lesson 2 Exit Ticket lets students assess their comprehension of synthetic division. Students must demonstrate their comprehension of the method by matching the unique roots to each polynomial.  T: Group and partner work are used to facilitate student collaboration. The emphasis should be on communicating mathematical ideas using vocabulary phrases that are appropriate for the subject. Partnering exercises challenge students to represent their individual grasp of the idea while also evaluating the representations of other students. Student sharing of ideas and insights is an effective strategy to improve learning without your involvement.  Logical/Sequential/Visual Learners: Some students may find it beneficial to see numerous written representations of the distinct steps of synthetic division. When writing or pointing to terms and operations, use their names appropriately.  O: The lesson begins with a simple practice that demonstrates shortcuts and simplified processes, particularly in math. This leads nicely into the discussion of long and synthetic division. Concepts and vocabulary are explained, and students then practice their skills. The class returns for additional teaching and practice in pairs. This lesson combines whole-class, individual, and group work; one action flows into the next.

Procedures

After this lesson, students will understand what the factor theorem is and how it relates roots to factors. They will learn how synthetic division indicates whether a value is a root of a polynomial, and how synthetic division gives the same results as long division while being considerably easier and faster. This lesson expands on the fundamentals of polynomials and presents aspects of their graph. Students must comprehend where roots come from and how to use polynomial functions in real-world situations. Students will be able to utilize synthetic division to determine whether a given value is a root. They will be able to write polynomial equations with known roots as well as factorized polynomials. Ask students to examine a parabola's graph, such as this one: y = x² + 2x - 3. Then have students respond to the following questions: "What is the degree of the graph, and how many x-intercepts are there? (Refer to the fundamental theorem of algebra.)" "What are the x-intercepts and zeros?" "What is the factored form of the parabola's equation?" "What is the relationship between zeros and factors?" “Since x² + 2x − 3 = (x + 3)(x − 1), that means \((x^2 + 2x - 3)\over (x + 3)\) = x - 1. We use polynomial division to find the roots of polynomials. How do we divide polynomials? We can apply what we know from long division, or we can skip forward and use synthetic division." Long Division:<figure class="image image_resized" style="width:49.89%;"><img style="aspect-ratio:282/119;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_46.png" width="282" height="119"></figure>Step 1: Divide x² by x to get x (line up like terms)  (x² / x) = x  Step 2: Multiply x by x + 3 and record the result beneath the dividend.  Step 3: Subtract the product by distributing the negative sign across all its terms.  Step 4: -x divided by x = -1 or (-x / x) = -1.  Step 5: Multiply -1 by x + 3 and write the product below; then subtract.<figure class="image image_resized" style="width:47.72%;"><img style="aspect-ratio:455/311;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_47.png" width="455" height="311"></figure>Step 5 Second Example of Long Division:<figure class="image image_resized" style="width:20.92%;"><img style="aspect-ratio:180/188;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_48.png" width="180" height="188"></figure>"This procedure resembles long division. We will divide polynomials using a simpler approach known as synthetic division." "Synthetic division requires that the divisor be of the form x − a or x + a. Using the same example as before, divide x² + 2x - 3 by x + 3." The notes below should be posted on the board for students to copy. Step 1: Write the dividend coefficients (1, 2, −3) and draw a "L" around the numbers and a dashed line in front of the constant (optional). Step 2: Since the divisor is x + 3, and x + 3 = 0, and x = -3, we'll pick -3 for synthetic division. Place -3 to the left of the 1 on the other side of the line.<figure class="image image_resized" style="width:42.29%;"><img style="aspect-ratio:385/144;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_49.png" width="385" height="144"></figure>Step 3: Bring down the 1. (Always bring down the first coefficient.)  Step 4: Multiply the 1 by the divisor - 3 and insert the product under the 2.<figure class="image image_resized" style="width:51.48%;"><img style="aspect-ratio:392/108;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_50.png" width="392" height="108"></figure>Step 5: Sum 2 + −3 and write it under the line. Step 6: Multiply the − 1 by the divisor − 3 and put the result under the − 3.<figure class="image image_resized" style="width:52.94%;"><img style="aspect-ratio:444/109;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="444" height="109"></figure>Step 7: Add -3 + 3 and write the result below the line.  Step 8: Write down the quotient. The rest is the number to the right of the black line. The first integer to the left of the dashed line is the new constant, then comes the coefficient in front of x², and so on. This method works with polynomials of third, fourth, and higher degrees, as long as the divisor is of the type x ± a.<figure class="image image_resized" style="width:42.55%;"><img style="aspect-ratio:407/196;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_52.png" width="407" height="196"></figure>Third example for synthetic division:<figure class="image image_resized" style="width:25.02%;"><img style="aspect-ratio:171/48;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_53.png" width="171" height="48"></figure>Note: In Step 2, you need to explain that in synthetic division, you use the sign that is the opposite of the number. Don't forget that the constant term and its additive negative must add up to zero. It was x + 3, so in synthetic division, it should be -3. If we knew how to divide x - 4 by itself, we would use +4. At this point, pause and ask anyone if they have any questions. Activity 1: Group Work If the classroom has whiteboards on all four walls, write these division questions on them. Put the questions on poster paper and hang them up around the room if there aren't any whiteboards all the way around. Long Division: 1. (\(x^3\) + x² − 5x − 2) ÷ (x − 2) 2. (x² + 2x + 1) ÷ (x + 1) 3. (\(x^3\) − 3x² − 11x + 5) ÷ (x² + 2x − 1) 4. (\(x^4\) + 4\(x^3\) + 6x² + 4x + 1) ÷ (x + 1) Answers: 1. x² + 3x + 1 2. x + 1 3. (x – 5) 4. \(x^3\) + 3x² + 3x + 1 Synthetic Division: 5. (9\(x^3\) − 6x² − x + 30) ÷ (x − 3) 6. (x² − 3x − 28) ÷ (x + 4)   7. (\(x^3\) − x² − 2x) ÷ (x + 1) 8. (\(x^3\) + 7x² − 38x + 40) ÷ (x − 2) 9. (\(x^4\) + 3\(x^3\) − 25x² + 33x + 72) ÷ (x + 6) 10. (x² + x − 72) ÷ (x − 8)    Answers: 5. (9x² + 21x + 62) + (216 / x – 3) Answers: 6. (x – 7) Answers: 7. (\(x^3\) – 2x) Answers: 8. (x² + 9x – 20) Answers: 9. (\(x^3\) – 3x² – 7x + 75) – (378 / x + 6) Answers: 10. (x + 9) Divide the students into groups of three to four, and assign each group to a problem on the board. Allow students to work on the problem in front of them for one to two minutes. If they finish the task in that time, they should raise their hands and ask you to check their work. If they are correct, they should erase their work so that the following group does not see it. If they do not finish the problem, it is fine. When the time is up, have the class rotate clockwise. The groups must then look at the next challenge and determine if the group in front of them was on the correct track in solving it. The current group completes the problem and has the work checked. Repeat the process for 10 to 15 minutes, ensuring that each group has seen at least one long division problem and four to five synthetic division questions. Bring the entire class back together. Ask, "Which process did you like better and why?" They will most likely want to know why synthetic division works. Now is an excellent time to introduce the Factor Theorem. "Determining the x-intercepts is crucial for information about the real-world scenario, as polynomials can be utilized to model real-world applications." Post the following on the board for students to copy into their notes. Factor theorem: If a is used to synthetically divide a polynomial and it produces a remainder of zero, then not only is x = a a root of the polynomial, but x − a is a factor of the polynomial. In the previous example, −3 was used in a synthetic division with the polynomial x² + 2x − 3, resulting in a remainder of 0. Therefore, x − (−3) or x + 3 is a factor of the polynomial. "Synthetic division is a step-by-step procedure in which each stage provides knowledge essential for the subsequent phase. In this sense, it builds on itself." Example: Find the roots of the polynomial \(x^3\) + 5x² − x − 5: 1, − 1, 2, and − 5.<figure class="image image_resized" style="width:49.53%;"><img style="aspect-ratio:477/348;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_54.png" width="477" height="348"></figure>Example: (\(x^3\) – 5x – 4) ÷ (x – 3) 3     1     0     –5     –4               3      9     12        1     3      4     8 Answer: x² + 3x + 4 + ( 8 / x – 3 ) Activity 2: Pairs After projecting the following table onto the board, have a class discussion about the middle column.<figure class="image"><img style="aspect-ratio:519/368;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_55.png" width="519" height="368"></figure>“You and your partner will determine which numbers are and are not roots. You should utilize synthetic division, as shown in the previous example. In the second polynomial, you must use a 0 coefficient for the \(x^3\) and x terms.”<figure class="image"><img style="aspect-ratio:475/597;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_57.png" width="475" height="597"></figure>Walk around and check the students' work. Allow students who are catching on to rapidly write the polynomials in factored form. In the previous example, if −1, 1, and −5 are roots of the polynomial \(x^3\) + 5x² − x − 5, they can write the roots as factors: \(x^3\) + 5x² − x − 5 = (x + 1)(x − 1)(x + 5). Choose a few groups to write their work on the whiteboard. This is an excellent opportunity for students to review their work and ask questions. Activity 3: "To create your own polynomials, take these steps: Consider three to five numbers to represent the roots of a polynomial. Write the roots in factored form. Multiply all of the elements together to generate a standard polynomial." Next, have students deliver their polynomial to the student behind them, along with two roots and a non-root number. Students must then identify the roots and non-roots. When everyone is done, students return their work to the person in front of them to check for faults. Another example: (x + 7)(x – 6)(x – 11) \(x^3\) – 10x² – 53x + 462 possible roots, –2, –1, 1, 2 Use the Lesson 2 Exit Ticket (M-A2-3-2_Lesson 2 Exit Ticket and KEY in the Resources folder) to assess students' comprehension. Use synthetic division to connect the polynomials on the left to their roots on the right. Extension: Determine the other binomial factor of the following trinomials if x = a is a root of each polynomial. 1. x² – ax + 3x – 3a                (x + 3) 2. x² – ax – 5x + 5a                (x – 5) 3. x² – ax + bx – ab                (x + b) 4. x² – ax – bcx + abc             (x – bc) 5. 6\(x^3\) – x² – 13x + 3            (2x + 3)

Layla Nguyen

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Synthetic Division and the Factor Theorem (M-A2-3-2)

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Layla Nguyen

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Description

This lesson will build on the topic of polynomials and their roots. Students will:
- show how roots and factors are related.
- determine whether a given value is a root of a polynomial using synthetic division.

Grade

Grade 11 Grade 12