Proves Remainder Theorem used for COT.pptx

Design INteractive IMs

1. Write each polynomial in descending order of x and give its degree. 1. x 3 + x 2 – 22x - 25x 5 + 2 2. 4x 2 + 21x 5 - 26x 3 + 28x - 10 + 5x 4 3. 6 – 31x + 3x 3 – 2x 4 4. x 3 + 7x 2 + 5x 4 – 25x + 5 5. x 3 + 7x 2 + 5 – 25x + 5x 5  

2. Guide Questions: 1. How did you arrange the following polynomials? 2. In your own opinion, is it necessary to arrange the polynomials before performing division? Why? 3. In real-life scenario, how will you consider dividing toxic or negative thoughts in our body system?

3. ACTIVITY No. 1 (GROUP TASK) Group 1 (Using long division solve the given polynomials (30x 5 – 50x 4 – 21x 2 + 32x - 8) ÷ (3x - 5), show your complete solution in 1 manila paper)   Group 2 (Using synthetic division solve the given polynomials (6x 3 + 11x 2 – 4x – 9) ÷ (x + 2), show your complete solution in 1 manila paper)   Group 3 (Construct a Venn Diagram and identify the differences and similarities of using long division and synthetic division.) Group 4 (Conduct a survey within a group of which is the most preferable technique to be used in dividing polynomials) Group 5 (Create a poem about diversity.) Group 6 (Solve the following polynomials by dividing by using any method to find the hidden words)

4. Guide Questions 1. What do you think is the consideration to use long and synthetic division? 2. What do you think are the common characteristics of long and synthetic division? 3. How did you find the activity? 4. What realizations do you have after the activity?

5. Proves the Remainder Theorem and the Factor Theorem

6. OBJECTIVES find the remainder using the Remainder Theorem determines whether (x – r) is a factor of a given polynomial develops patience on how to solve exercises in remainder theorem

7. Activity: Directions: Fill in the blanks with words and symbols that will best complete the statements given below. Suppose that the polynomial P(x) is divided by (x – r), as follows: If P(x) is of degree n, then Q(x) is of degree _____. The remainder R is a constant because ____________________. Now supply the reasons for each statement in the following table.

8. The Remainder Theorem If the polynomial P(x) is divided by (x – r), the remainder R is a constant and is equal to P(r). R = P(r) Thus, there are two ways to find the remainder when P(x) is divided by (x – r), that is: (1) use synthetic division, or (2) calculate P(r) Similarly, there are two ways to find the value of P(r): (1) substitute r in the polynomial expression P(x), or (2) use synthetic division.

9. Illustrative Example 1 Example 1. Find the remainder when (5x2 – 2x + 1) is divided by (x + 2). Solution:

10. Using the Remainder Theorem: P(x) = 5x 2 – 2x + 1, r = –2 P(–2) = 5(–2) 2 – 2(–2) + 1 P(–2) = 5(4) + 4 + 1 P(–2) = 20 + 4 + 1 = 25 Therefore, the remainder when P(x) = 5x 2 – 2x + 1 is divided by x + 2 is 25.

11. Use the Remainder Theorem to find the remainder when the given polynomial is divided by each binomial. Verify your answer using synthetic division. Indicate whether or not each binomial is a factor of the given polynomial. 1. P(x) = x 3 – 7x + 5 a. x – 1 b. x + 1 c. x – 2 2. P(x) = 2x 3 – 7x + 3 a. x – 1 b. x + 1 c. x – 2

12. Written Work Use the Remainder Theorem to find the remainder R in each of the following. 1. P(x) = 4x 3 – 3x 3 – x 2 + 2x + 1 a. x – 1 b. x + 1 c. x – 2 2. P(x) = 2x 4 – 3x 3 + 4x 2 + 17x + 7 a. 2x – 3 b. 2x + 3 c. 3x – 2

13. Generalization/Abstraction Labarn o Labarn na Labarn

14. Performance Task/Evaluation Use the Remainder Theorem to find the remainder R in each of the following. 1. (x 4 – x 2 + 2) ÷ (x + 2) 2. (x 3 – 2x 2 + x + 6) ÷ (x – 3) 3. (x 4 – 3x 3 + 4x 2 – 6x + 4) ÷ (x – 2) 4. (x 4 – 16x 3 + 18x 2 – 128) ÷ (x + 2) 5. (3x 2 + 5x 3 – 8) ÷ (x – 4)

15. Success is very much the intersection of luck and hard work Thank You for Great Participation