1. Splash Screen

2. Five-Minute Check (over Chapter 4) CCSS Then/Now New Vocabulary
Concept Summary: Properties of Exponents
Key Concept: Simplifying Monomials
Example 1: Simplify Expressions
Example 2: Degree of a Polynomial
Example 3: Simplify Polynomial Expressions
Example 4: Simplify by Using the Distributive Property
Example 5: Real-World Example: Write a Polynomial Expression
Example 6: Multiply Polynomials
Lesson Menu

3. Find the x-coordinate of the vertex of f(x) = 5x 2 + 15x + 1.B. C.
D. –3
5-Minute Check 1

4. Find the x-coordinate of the vertex of f(x) = 5x 2 + 15x + 1.B. C.
D. –3
5-Minute Check 1

5. Write a quadratic equation with roots of –3 and
Write a quadratic equation with roots of –3 and Write the equation in the form ax 2 + bx + c = 0, where a, b, and c are integers.
A. 12x 2 + 4x – 3 = 0
B. 4x 2 + 9x – 9 = 0
C. 3x 2 + 7x – 4 = 0
D. –x 2 + 4x + 3 = 0
5-Minute Check 2

6. Write a quadratic equation with roots of –3 and
Write a quadratic equation with roots of –3 and Write the equation in the form ax 2 + bx + c = 0, where a, b, and c are integers.
A. 12x 2 + 4x – 3 = 0
B. 4x 2 + 9x – 9 = 0
C. 3x 2 + 7x – 4 = 0
D. –x 2 + 4x + 3 = 0
5-Minute Check 2

7. Find the exact solutions for –x 2 + 8x – 12 = 0 using the method of your choice.
B. 1, 3
C. –1, 2
D. –3, 2
5-Minute Check 3

8. Find the exact solutions for –x 2 + 8x – 12 = 0 using the method of your choice.
B. 1, 3
C. –1, 2
D. –3, 2
5-Minute Check 3

9. Find the number of seconds it will take an object to land on the ground if it is dropped from a height of 300 feet, assuming there is no air resistance. Use the equation h(t) = –16t 2 + h0, where h(t) is the height of the object in feet at the time t, t is the time in seconds, and h0 is the initial height in feet. Round to the nearest tenth, if necessary.
A s
B s
C. 7.4 s
D. 4.3 s
5-Minute Check 4

10. Find the number of seconds it will take an object to land on the ground if it is dropped from a height of 300 feet, assuming there is no air resistance. Use the equation h(t) = –16t 2 + h0, where h(t) is the height of the object in feet at the time t, t is the time in seconds, and h0 is the initial height in feet. Round to the nearest tenth, if necessary.
A s
B s
C. 7.4 s
D. 4.3 s
5-Minute Check 4

11. Find the solution to the quadratic inequality x 2 + 3x ≤ 54.
A. x ≤ –9
B. –9 ≤ x ≤ 6
C. x ≥ 6
D. x ≤ 54
5-Minute Check 5

12. Find the solution to the quadratic inequality x 2 + 3x ≤ 54.
A. x ≤ –9
B. –9 ≤ x ≤ 6
C. x ≥ 6
D. x ≤ 54
5-Minute Check 5

13. Mathematical Practices 2 Reason abstractly and quantitatively.
Content Standards
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.
Mathematical Practices
2 Reason abstractly and quantitatively.
CCSS

14. Add, subtract, and multiply polynomials.
You evaluated powers.
Multiply, divide, and simplify monomials and expressions involving powers.
Add, subtract, and multiply polynomials.
Then/Now

15. simplify
degree of a polynomial
Vocabulary

16. Concept

17. Concept

18. A. Simplify the expression . Assume that no variable equals 0.
Simplify Expressions
A. Simplify the expression Assume that no variable equals 0.
Original expression
Definition of negative exponents
Definition of exponents
Example 1

19. Divide out common factors.
Simplify Expressions
Divide out common factors.
Simplify.
Example 1

20. Divide out common factors.
Simplify Expressions
Divide out common factors.
Simplify.
Example 1

21. B. Simplify the expression . Assume that no variable equals 0.
Simplify Expressions
B. Simplify the expression Assume that no variable equals 0.
Quotient of powers
Subtract powers.
Definition of negative exponents
Answer:
Example 1

22. B. Simplify the expression . Assume that no variable equals 0.
Simplify Expressions
B. Simplify the expression Assume that no variable equals 0.
Quotient of powers
Subtract powers.
Definition of negative exponents
Answer:
Example 1

23. C. Simplify the expression . Assume that no variable equals 0.
Simplify Expressions
C. Simplify the expression Assume that no variable equals 0.
Power of a quotient
Power of a product
Example 1

24. Simplify Expressions
Power of a power
Example 1

25. Simplify Expressions
Power of a power
Example 1

26. A. Simplify the expression . Assume that no variable equals 0.C. D.
Example 1

27. A. Simplify the expression . Assume that no variable equals 0.C. D.
Example 1

28. B. Simplify the expression Assume that no variable equals 0.C. D.
Example 1

29. B. Simplify the expression Assume that no variable equals 0.C. D.
Example 1

30. C. Simplify the expression . Assume that no variable equals 0.D.
Example 1

31. C. Simplify the expression . Assume that no variable equals 0.D.
Example 1

32. Degree of a Polynomial
Answer:
Example 2

33. Degree of a Polynomial
Answer:
Example 2

34. Degree of a Polynomial
Answer:
Example 2

35. Degree of a Polynomial
Answer: This expression is a polynomial because each term is a monomial. The degree of the first term is 5 and the degree of the second term is or 9. The degree of the polynomial is 9.
Example 2

36. Degree of a Polynomial
C. Determine whether is a polynomial. If it is a polynomial, state the degree of the polynomial.
Answer:
Example 2

37. Degree of a Polynomial
C. Determine whether is a polynomial. If it is a polynomial, state the degree of the polynomial.
Answer: The expression is not a polynomial because is not a monomial: Monomials cannot contain variables in the denominator.
Example 2

38. A. Is a polynomial? If it is a polynomial, state the degree of the polynomial.
A. yes, 5
B. yes, 8
C. yes, 3
D. no
Example 2

39. A. Is a polynomial? If it is a polynomial, state the degree of the polynomial.
A. yes, 5
B. yes, 8
C. yes, 3
D. no
Example 2

40. B. Is a polynomial? If it is a polynomial, state the degree of the polynomial.
A. yes, 2
B. yes,
C. yes, 1
D. no
Example 2

41. B. Is a polynomial? If it is a polynomial, state the degree of the polynomial.
A. yes, 2
B. yes,
C. yes, 1
D. no
Example 2

42. C. Is a polynomial? If it is a polynomial, state the degree of the polynomial.
A. yes, 5
B. yes, 6
C. yes, 7
D. no
Example 2

43. C. Is a polynomial? If it is a polynomial, state the degree of the polynomial.
A. yes, 5
B. yes, 6
C. yes, 7
D. no
Example 2

44. A. Simplify (2a3 + 5a – 7) – (a3 – 3a + 2).
Simplify Polynomial Expressions
A. Simplify (2a3 + 5a – 7) – (a3 – 3a + 2).
(2a3 + 5a – 7) – (a3 – 3a + 2)
Distribute the –1.
Group like terms.
= a3 + 8a – 9 Combine like terms.
Answer:
Example 3

45. A. Simplify (2a3 + 5a – 7) – (a3 – 3a + 2).
Simplify Polynomial Expressions
A. Simplify (2a3 + 5a – 7) – (a3 – 3a + 2).
(2a3 + 5a – 7) – (a3 – 3a + 2)
Distribute the –1.
Group like terms.
= a3 + 8a – 9 Combine like terms.
Answer: a3 + 8a – 9
Example 3

46. B. Simplify (4x2 – 9x + 3) + (–2x2 – 5x – 6).
Simplify Polynomial Expressions
B. Simplify (4x2 – 9x + 3) + (–2x2 – 5x – 6).
Align like terms vertically and add.
4x2 – 9x + 3
(+) –2x2 – 5x – 6
2x2 – 14x – 3
Answer:
Example 3

47. B. Simplify (4x2 – 9x + 3) + (–2x2 – 5x – 6).
Simplify Polynomial Expressions
B. Simplify (4x2 – 9x + 3) + (–2x2 – 5x – 6).
Align like terms vertically and add.
4x2 – 9x + 3
(+) –2x2 – 5x – 6
2x2 – 14x – 3
Answer: 2x2 – 14x – 3
Example 3

48. A. Simplify (3x2 + 2x – 3) – (4x2 + x – 5).
A. 7x 2 + 3x – 8
B. –x 2 + 3x – 8
C. –x 2 + 3x + 2
D. –x 2 + x + 2
Example 3

49. A. Simplify (3x2 + 2x – 3) – (4x2 + x – 5).
A. 7x 2 + 3x – 8
B. –x 2 + 3x – 8
C. –x 2 + 3x + 2
D. –x 2 + x + 2
Example 3

50. B. Simplify (–3x2 – 4x + 1) – (4x2 + x – 5).
A. 9x 2 + 6x + 7
B. –7x 2 – 5x + 6
C. 3x 2 – 6x + 7
D. 3x 2 – 2x + 6
Example 3

51. B. Simplify (–3x2 – 4x + 1) – (4x2 + x – 5).
A. 9x 2 + 6x + 7
B. –7x 2 – 5x + 6
C. 3x 2 – 6x + 7
D. 3x 2 – 2x + 6
Example 3

52. = –y(4y2) – y(2y) – y(–3) Distributive Property
Simplify by Using the Distributive Property
Find –y(4y2 + 2y – 3).
–y(4y2 + 2y – 3)
= –y(4y2) – y(2y) – y(–3) Distributive Property
= –4y3 – 2y2 + 3y Multiply the monomials.
Answer:
Example 4

53. = –y(4y2) – y(2y) – y(–3) Distributive Property
Simplify by Using the Distributive Property
Find –y(4y2 + 2y – 3).
–y(4y2 + 2y – 3)
= –y(4y2) – y(2y) – y(–3) Distributive Property
= –4y3 – 2y2 + 3y Multiply the monomials.
Answer: –4y3 – 2y2 + 3y
Example 4

54. Find –x(3x3 – 2x + 5). A. –3x2 – 2x + 5 B. –4x2 – 3x2 – 6x
C. –3x4 + 2x2 – 5x
D. –3x4 – 2x3 + 5x
Example 4

55. Find –x(3x3 – 2x + 5). A. –3x2 – 2x + 5 B. –4x2 – 3x2 – 6x
C. –3x4 + 2x2 – 5x
D. –3x4 – 2x3 + 5x
Example 4

56. Write a Polynomial Expression
E-SALES A small online retailer estimates that the cost, in dollars, associated with selling x units of a particular product is given by the expression 0.001x 2 + 5x The revenue from selling x units is given by 10x. Write a polynomial to represent the profits generated by the product if profit = revenue – cost.
Example 5

57. 10x – (0.001x 2 + 5x + 500) Original expression
Write a Polynomial Expression
10x – (0.001x 2 + 5x + 500) Original expression
= 10x – 0.001x 2 – 5x – 500 Distributive Property
= –0.001x 2 + 5x – 500 Add 10x to –5x.
Answer:
Example 5

58. 10x – (0.001x 2 + 5x + 500) Original expression
Write a Polynomial Expression
10x – (0.001x 2 + 5x + 500) Original expression
= 10x – 0.001x 2 – 5x – 500 Distributive Property
= –0.001x 2 + 5x – 500 Add 10x to –5x.
Answer: The polynomial is –0.001x 2 + 5x – 500.
Example 5

59. INTEREST Olivia has $1300 to invest in a government bond that has an annual interest rate of 2.2%, and a savings account that pays 1.9% per year. Write a polynomial for the interest she will earn in one year if she invests x dollars in the government bond.
A. –0.003x
B. – 0.003x
C x
D x
Example 5

60. INTEREST Olivia has $1300 to invest in a government bond that has an annual interest rate of 2.2%, and a savings account that pays 1.9% per year. Write a polynomial for the interest she will earn in one year if she invests x dollars in the government bond.
A. –0.003x
B. – 0.003x
C x
D x
Example 5

61. Distributive Property
Multiply Polynomials
Find (a2 + 3a – 4)(a + 2).
(a2 + 3a – 4)(a + 2)
Distributive Property
Distributive Property
Multiply monomials.
Example 6

62. = a3 + 5a2 + 2a – 8 Combine like terms.
Multiply Polynomials
= a3 + 5a2 + 2a – 8 Combine like terms.
Answer:
Example 6

63. = a3 + 5a2 + 2a – 8 Combine like terms.
Multiply Polynomials
= a3 + 5a2 + 2a – 8 Combine like terms.
Answer: a3 + 5a2 + 2a – 8
Example 6

64. Find (x2 + 3x – 2)(x + 4). A. x 3 + 7x 2 + 10x – 8 B. x 2 + 4x + 2
C. x 3 + 3x 2 – 2x + 8
D. x 3 + 7x x – 8
Example 6

65. Find (x2 + 3x – 2)(x + 4). A. x 3 + 7x 2 + 10x – 8 B. x 2 + 4x + 2
C. x 3 + 3x 2 – 2x + 8
D. x 3 + 7x x – 8
Example 6

66. End of the Lesson