Adding, Subtracting, Multiplying, and Factoring Polynomials

1. - - - x x x + + + Adding, Subtracting, Multiplying, and Factoring Polynomials Click on a type above to review. (5x2 + 2x + 5) + (7x2 - 7x + 8) (3x2 + 10x – 28) + (4x2 + 12x – 15) (x+4)(3x - 1) 3x4(x + 2)

2. Adding Polynomials • Add like terms together • Write in standard form

3. Subtracting Polynomials • Change subtraction to add the opposite • Follow adding rules

4. Multiplying Polynomials • Use FOIL when multiplying two binomials. • Use the Box Method.

5. Like Terms Terms with the same variable and same exponent. Which of the following terms are like terms? 2x and -13x 2x2 and -13x 8y4 and -8x4 x3 and x2

6. Like Terms Terms with the same variable and same exponent. Which of the following terms are like terms? 2x2 and -13x These are not terms because, although they have the same variable, the exponents are different.

7. Like Terms Terms with the same variable and same exponent. Which of the following terms are like terms? x3 and x2 These are not terms because, although they have the same variable, the exponents are different.

8. Like Terms Terms with the same variable and same exponent. Which of the following terms are like terms? 8y4 and -8x4 These are not terms because they do not have the same variable.

9. Standard Form of a Polynomial Polynomial is written in standard form when the terms are in order from greatest exponent to least exponent. Example: 5x3 + 10x2 + 6x - 8 is in standard form. -10x2 + 7x + 4x3 + 3 is not in standard form.

10. Add the Opposite • To add the opposite, change subtraction to addition and add the opposite of the term being subtracted. • If you are subtracting a polynomial in parentheses, change everything the second parentheses to the opposite sign. - + +

11. FOIL Method (x+7)(x - 4) • Multiply the FIRST terms in each binomial • Multiply the OUTER terms in each binomial • Multiply theINNER terms in each binomial • Multiply the LAST terms in each binomial • Add like terms and write in standard form (x+7)(x + -4) (x+7)(x + -4) (x+7)(x + -4) (x+7)(x + -4)

12. Box Method (x+7)(x - 4) • Draw a box. • Write one factor on one side and write the other factor on the other side. • Find the area of each small box. • Add like terms of the small box areas, and write in standard form. x 7 x x2 7x -28 -4 -4x x2 + 7x + -4x + -28 ( x+7)(x-4)=x2 + 3x + -28

13. Factoring Polynomials Click on a form to review how to factor x2+bx+c form ax2+bx+c form

14. Factoring x2 + bx + c form • Find two factors of c whose sum is b. • Example: To factor x2 + 7x + 12, find factors of 12 whose sum is 7: (choose one below) a. 6 and 2 b. -3 and -4 c. 3 and 4

15. Oops! Nice try! You found factors of 12, but their sum is not 7. Try again!

16. A binomial is a polynomial with two terms. Which of the following are binomials? 12x2 4x + 1 7x2 - 3x + 2 -14x4 - 2

17. A binomial is a polynomial with two terms. Which of the following are binomials? 12x2 This is a monomial because there is only one thing being added.

18. A binomial is a polynomial with two terms. Which of the following are binomials? 7x2 - 3x + 2 This is a trinomial because there are three things being added.

19. Great job! 3 and 4 are factors of 12, and they add up to 7.

20. Factoring x2 + bx + c formFactoring x2 + 7x + 12 Since 3 and 4 are the factors of 12, the factors of the polynomial are (x + 4) and (x + 3) So, x2 + 7x + 12 = (x + 4)(x + 3)

21. Factoring ax2+bx+c form • Multiply the first and last terms. • Find factors of that term whose sum is bx. Example: To factor 2x2+11x+5, multiply 2x2 and 5 to get 10x2. Then find factors of 10x2 whose sum is 11x. (Click on the correct factors below.) a. 5x and 2x b. 10x and x c. -10x and -1x

22. Oops! Nice try! You found factors of 10x2, but their sum is not 11x. Try again!

23. Great job! 10x and x are factors of 10x2, and they add up to 11x.Go on to the next step.

24. Factoring 2x2+11x+5 • Now that you’ve found the two factors,10x and x, make a box. • Put the first term of the polynomial in the first square • Put the last term in the polynomial in the last square • Put the two factors in the 2nd and 3rd square. 2x2 10x x 5

25. Factoring 2x2+11x+5 • Now factor out the common factor in each row and column. • The factors are the sides of the box. 2x2+11x+5 = (x + 5)(2x + 1) x 5 2x 2x2 10x x 5 1 Now try one on your own

26. Factor 3x2 + 10x + 8 Which of the following are the factors? a. (3x + 2)(x + 4) b. (3x + 8)(x +1) c. (3x + 4)(x + 2)

27. You are ready to move on to the maze! I love algebra!

28. Pick a path to begin from. Path 1 Or a. 5x2 - 2x + 5 b. 5x2 + 8x + 5 c. 5x2 + 2x + 5 Path 2 a. 5x2+2x+4 b. 5x2-2x+4 c. 5x2+6x+4

29. a. x2 - 7x + 6 b. x2 + 7x + 6 c. 7x2 - 7x + 8

30. a. -3x2 - 2x+5 b. -3x2 - 2x + 7 c. -3x2 +8x - 7

31. a. x2 - 2x - 6 b. x2 + 6x - 6 c. x2 - 6x + 4

32. a. 3x2 + 10x - 28 b. 2x2 + 10x - 28 c. 3x2 + 18x - 28

33. a. 4x2 + 12x - 15 b. 3x2 + 12x + 15 c. 3x2 - 12x - 15

34. a. 5x3 + 3x2 - 4x b. 5x3 + 10x2 + 6x - 8 c. 5x3 + 13x2 + 2x - 8

35. a. 4x3 - 2x2 - 3 b. 4x3+10x2 -5x-3 c. 4x3+10x2 - 7x -3

36. a. -5x6 +10x4-35x4 b. 5x6+10x5-35x4 c. -5x6+10x5-35x4

37. Go back and try again!

38. a. -3x7-12x5-6x2 b. 3x7-12x5-6x2 c. 3x7-x5-6x2

39. a. 49m2 - 14m - 4 b. 49m2 - 28m + 4 c. 49m2-14m+4

40. a. 16w2-20w-25 b. 16w2 - 40w + 25 c. 16w2 - 40w - 25

41. a. 2, 3 b. 3, -2 c. -3, -2

42. a. 3, -7 b. 7, -3 c. -7, -3

43. a. (x+3)(x+2) b. (x-3)(x-2) c. (x-3)(x+2)

44. a. (x - 7)(x + 3) b. (x - 7)(x - 3) c. (x + 7)(x - 3)

45. a. (x-7)(x-5) b. (x+7)(x-5) c. (x-7)(x+5)

46. a. (x + 6)(x - 1) b. (x - 3)(x - 2) c. (x + 3)(x + 2)

47. a. (4x+3)(x+1) b. (4x-1)(3x-1) c. (4x+1)(3x+1)

48. OOPS! Go back and try again!

49. a. -3, 3 b. 3 c. 9, 4

50. a. -2, 2 b. -2 c. -6, 6