Dividing Polynomials 9-12-’12

1. Dividing Polynomials9-12-’12

2. Factoring Polynomials, Roots of Real Numbers 9-13-’12

3. CCSS:N.RN.2 & A.APR.1 N.RN.2 REWRITE expressions involving radicals and rational exponents using the properties of exponents A.APR.1 UNDERSTAND that polynomials FORM a system analogous to the integers, namely, they ARE CLOSED under the operations of addition, subtraction, and multiplication; ADD, SUBTRACT, and MULTIPLY polynomials.

4. Essential Question(s): • How do I divide polynomial expressions? • How do I factor a polynomial expression? • How do I interpret the parts of a factored expression in context of the variables? • How do I evaluate roots of real numbers? • How do I rewrite roots of real numbers as rational exponents?

5. Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2. Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6. Attend to precision. • 7. Look for and make use of structure. • 8. Look for and express regularity in repeated reasoning.

6. Simple Division - dividing a polynomial by a monomial

7. Simplify

8. Simplify

9. Long Division - divide a polynomial by a polynomial • Think back to long division from 3rd grade. • How many times does the divisor go into the dividend? Put that number on top. • Multiply that number by the divisor and put the result under the dividend. • Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.

10. x2/x = x -8x/x = -8 x - 8 + 3x -( ) x2 - 8x - 24 -( ) - 8x - 24 0

11. h3/h = h2 4h2/h = 4h 5h/h = 5 h2 + 4h + 5 -( ) - 4h2 h3 - 11h 4h2 -( ) 4h2 - 16h 5h + 28 -( ) 5h - 20 48

12. Synthetic Division - divide a polynomial by a polynomial • To use synthetic division: • There must be a coefficient for every possible power of the variable. • The divisor must have a leading coefficient of 1.

13. Since the numerator does not contain all the powers of x, you must include a 0 for the Step #1: Write the terms of the polynomial so the degrees are in descending order.

14. 5 0 -4 1 6 Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients. Since the divisor is x-3, r=3

15. 5 Step #3: Bring down the first coefficient, 5.

16. Step #4: Multiply the first coefficient by r, so and place under the second coefficient then add. 15 5 15

17. 15 45 15 5 Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add. 41

18. 15 45 123 372 15 41 5 Step #5 cont.: Repeat the same procedure. Where did 123 and 372 come from? 124 378

19. 15 45 123 372 15 41 124 378 5 Step #6: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

20. The quotient is: Remember to place the remainder over the divisor.

21. Ex 7: Step#1: Powers are all accounted for and in descending order. Step#2: Identify r in the divisor. Since the divisor is x+4, r=-4 .

22. Step#3: Bring down the 1st coefficient. Step#4: Multiply and add. Step#5: Repeat. 4 -4 20 0 8 -1 1 0 -2 10 -5

23. Ex 8: Notice the leading coefficient of the divisor is 2 not 1. We must divide everything by 2 to change the coefficient to a 1.

24. 3

25. *Remember we cannot have complex fractions - we must simplify.

26. Ex 9: Coefficients 1

28. Factoring Polynomials

29. A “Difference of Squares” is a binomial (*2 terms only*) and it factors like this:

30. Factoring a polynomial means expressing it as a product of other polynomials.

31. Factoring Method #1 Factoring polynomials with a common monomial factor (using GCF). **Always look for a GCF before using any other factoring method.

32. Steps: 1. Find the greatest common factor (GCF). 2. Divide the polynomial by the GCF. The quotient is the other factor. 3. Express the polynomial as the product of the quotient and the GCF.

33. Step 1: Step 2: Divide by GCF

34. The answer should look like this:

35. Factor these on your own looking for a GCF.

36. Factoring Method #2 Factoring polynomials that are a difference of squares.

37. To factor, express each term as a square of a monomial then apply the rule...

38. Here is another example:

39. Try these on your own:

40. Sum and Difference of Cubes:

41. Apply the rule for sum of cubes: Write each monomial as a cube and apply either of the rules. Rewrite as cubes

42. Apply the rule for difference of cubes: Rewrite as cubes

43. Factoring a trinomial in the form: Factoring Method #3

44. Factoring a trinomial: 1. Write two sets of parenthesis, ( )( ). These will be the factors of the trinomial. 2. Product of first terms of both binomials must equal first term of the trinomial. Next

45. Factoring a trinomial: 3. The product of last terms of both binomials must equal last term of the trinomial (c). 4. Think of the FOIL method of multiplying binomials, the sum of the outer and the inner products must equal the middle term (bx).

46. x x -2 -4 O + I = bx ? 1x + 8x = 9x 2x + 4x = 6x -1x - 8x = -9x -2x - 4x = -6x Factors of +8: 1 & 8 2 & 4 -1 & -8 -2 & -4

47. F O I L Check your answer by using FOIL

48. Lets do another example: Don’t Forget Method #1. Always check for GCF before you do anything else. Find a GCF Factor trinomial

49. Step 1: When a>1 and c<1, there may be more combinations to try!

50. Step 2: Order can make a difference!