Division of Polynomials in Grade 10.pptx

PPt

1. Quarter 1: Week 10 December 9, 2020

2. Opening Prayer San Jose del Monte National High School Brgy . Yakal , Francisco Homes, CSJDM, Bulacan

3. San Jose del Monte National High School Brgy. Yakal, Francisco Homes, CSJDM, Bulacan 10- DAGOHOY

4. ANAGRAM GAME The object of the game is to rearrange or unscramble the letters to form a new word Let’s Play

5. A Ellipse Snip Oxymoron

6. Board! Activity

7. 7 ACTIVITY 1: SPOT THE DIFFERENCE Look at each pair of expressions below. Identify the expression that is not a polynomial from each. Write this expression on your whiteboard . A POLYNOMIAL EXPRESSION SHOULD NOT HAVE A VARIABLE IN THE DENOMINATOR

8. 8 Variables in a polynomial expression should not have negative exponents. Variables in a polynomial expression should not have fractional exponents

9. 9 Variables in a polynomial expression should not have fractional exponent Variables in a polynomial expression should not have a variable in the denominator equivalent to a variable with negative exponents

10. Some real-life situations require the application of polynomials. For example, engineers can use polynomials to create building plans and entrepreneurs can use polynomials to design cost-effective products .

11. Division of Polynomials Using Long and Synthetic Division

12. Objectives: Divide polynomials using long division and synthetic division Solve word problem that involves dividing polynomials

13. POLYNOMIAL EXPRESSION 13 is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non- negative integer exponents of variables.

14. 14 Example: 3x 3 + 4x 2 + x + 12

15. Examples Write the polynomials in standard form. Remember: The lead coefficient should be positive in standard form. To do this, multiply the polynomial by –1 using the distributive property.

16. Dividing Polynomials Dividing Polynomials Long division of polynomials is similar to long division of whole numbers. dividend = (quotient • divisor) + remainder The result is written in the form: quotient + When you divide two polynomials you can check the answer using the following:

17. Warm-Up Divide and Write 19 ÷ 5 = 2. 145 ÷ 11=

18. Divide using long division. Example 1: Using Long Division to Divide a Polynomial (– y 2 + 2 y 3 + 25) ÷ ( y – 3) 2 y 3 – y 2 + 0 y + 25 Step 1 Write the dividend in standard form, including terms with a coefficient of 0. Step 2 Write division in the same way you would when dividing numbers. y – 3 2y 3 – y 2 + 0 y + 25

19. Notice that y times 2y 2 is 2y 3 . Write 2y 2 above 2y 3 . Step 3 Divide. 2 y 2 –(2 y 3 – 6 y 2 ) Multiply y – 3 by 2y 2 . Then subtract. Bring down the next term. Divide 5y 2 by y. 5 y 2 + 0 y + 5 y –(5 y 2 – 15 y ) Multiply y – 3 by 5y. Then subtract. Bring down the next term. Divide 15y by y. 15 y + 25 –(15 y – 45) 70 Find the remainder. + 15 Multiply y – 3 by 15. Then subtract. Example 1 Continued y – 3 2y 3 – y 2 + 0 y + 25

20. Step 4 Write the final answer. Example 1 Continued – y 2 + 2 y 3 + 25 y – 3 = 2 y 2 + 5 y + 15 + 70 y – 3

21. Check It Out! Example 2 Divide using long division. (15 x 2 + 8 x – 12) ÷ (3 x + 1) 15 x 2 + 8 x – 12 Step 1 Write the dividend in standard form, including terms with a coefficient of 0. Step 2 Write division in the same way you would when dividing numbers. 3x + 1 15x 2 + 8x – 12

22. Check It Out! Example 2 Continued Notice that 3x times 5x is 15x 2 . Write 5x above 15x 2 . Step 3 Divide. 5 x –(15 x 2 + 5 x ) Multiply 3x + 1 by 5x. Then subtract. Bring down the next term. Divide 3x by 3x. 3 x – 12 + 1 –(3 x + 1) –13 Find the remainder. Multiply 3x + 1 by 1. Then subtract. 3x + 1 15x 2 + 8x – 12

23. Check It Out! Example 2 Continued Step 4 Write the final answer. 15 x 2 + 8 x – 12 3 x + 1 = 5 x + 1 – 13 3 x + 1

24. Exercises: Try this one! Divide using long division. ( x 2 + 5 x – 28) ÷ ( x – 3)

25. Solution: Divide. x –( x 2 – 3 x ) 8 x – 28 + 8 –(8 x – 24) –4 Exercises Continued x – 3 x 2 + 5x – 28 x 2 + 5 x – 28 x – 3 = x + 8 – 4 x – 3

26. Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients. For synthetic division to work, the polynomial must be written in standard form, using 0 and a coefficient for any missing terms, and the divisor must be in the form ( x – a ).

27. Divide using synthetic division. (3 x 4 – x 3 + 5 x – 1) ÷ ( x + 2) Step 1 Find a . Use 0 for the coefficient of x 2 . For (x + 2), a = –2. a = –2 Example 1: Using Synthetic Division to Divide by a Linear Binomial 3 – 1 0 5 –1 –2 Step 2 Write the coefficients and a in the synthetic division format.

28. Example 1 Continued Draw a box around the remainder, 45. 3 –1 0 5 –1 –2 Step 3 Bring down the first coefficient. Then multiply and add for each column. –6 3 45 Step 4 Write the quotient. 3 x 3 – 7 x 2 + 14 x – 23 + 45 x + 2 Write the remainder over the divisor. 46 –28 14 –23 14 –7

29. Check It Out! Example 2 Divide using synthetic division. (6 x 2 – 5 x – 6) ÷ ( x + 3) Step 1 Find a . Write the coefficients of 6x 2 – 5x – 6. For (x + 3), a = –3 . a = –3 –3 6 –5 –6 Step 2 Write the coefficients and a in the synthetic division format.

30. Check It Out! Example 2 Continued Draw a box around the remainder, 63. 6 –5 –6 –3 Step 3 Bring down the first coefficient. Then multiply and add for each column. –18 6 63 Step 4 Write the quotient. 6 x – 23 + 63 x + 3 Write the remainder over the divisor. –23 69

31. Divide using synthetic division. ( x 2 – 3 x – 18) ÷ ( x – 6) Exercises: Try this One!

32. Exercises Continued There is no remainder. 1 –3 –18 6 Solution: 6 1 Write the quotient. x + 3 18 3

33. What I Have Learned! Dividend Divisor Quotient Remainder 2x 4 – x - 36 2 0 0 -1 -36 X-2 -6 2x 3 + 4x 2 + 8x + 15

34. Assessment no. 6: Lesson Quiz 2. Divide by using synthetic division. ( x 3 – 3 x + 5) ÷ ( x + 2) 1. Divide by using long division. (8 x 3 + 6 x 2 + 7) ÷ ( x + 2) 8 x 2 – 10 x + 20 – 33 x + 2 x 2 – 2 x + 1 + 3 x + 2