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Properties of Exponents and Simplifying Polynomials

This lesson covers simplifying monomials, degree of polynomials, simplifying polynomial expressions, using the distributive property, and writing polynomial expressions in real-world examples.

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Properties of Exponents and Simplifying Polynomials

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  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. A. B. C. D.–3 Find the x-coordinate of the vertex of f(x) = 5x2 + 15x + 1. 5-Minute Check 1

  4. Write a quadratic equation with roots of –3 and .Write the equation in the form ax2 + bx + c = 0, where a, b, and c are integers. A. 12x2 + 4x – 3 = 0 B. 4x2 + 9x – 9 = 0 C. 3x2 + 7x – 4 = 0 D. –x2 + 4x + 3 = 0 5-Minute Check 2

  5. Find the exact solutions for –x2 + 8x – 12 = 0 using the method of your choice. A. 2, 6 B. 1, 3 C. –1, 2 D. –3, 2 5-Minute Check 3

  6. 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) = –16t2 + 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. 16.9 s B. 12.6 s C. 7.4 s D. 4.3 s 5-Minute Check 4

  7. Find the solution to the quadratic inequality x2 + 3x ≤ 54. A.x ≤ –9 B. –9 ≤ x ≤ 6 C.x ≥ 6 D.x ≤ 54 5-Minute Check 5

  8. 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

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

  10. simplify • degree of a polynomial Vocabulary

  11. Concept

  12. Concept

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

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

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

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

  17. Simplify Expressions Power of a power Example 1

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

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

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

  21. Answer: Degree of a Polynomial Example 2

  22. 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 2 + 7 or 9. The degree of the polynomial is 9. Example 2

  23. 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. Degree of a Polynomial Example 2

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

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

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

  31. 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

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

  33. 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 expression0.001x2 + 5x + 500. The revenue from sellingx units is given by 10x. Write a polynomial to represent the profits generated by the product if profit = revenue – cost. Example 5

  34. Write a Polynomial Expression 10x – (0.001x2 + 5x + 500) Original expression = 10x – 0.001x2 – 5x – 500 Distributive Property = –0.001x2 + 5x – 500 Add 10x to –5x. Answer:The polynomial is –0.001x2 + 5x – 500. Example 5

  35. 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 + 24.7 B. – 0.003x + 28.6 C. 0.003x + 28.6 D. 0.003x + 24.7 Example 5

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

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

  38. Find (x2 + 3x – 2)(x + 4). A.x3 + 7x2 + 10x – 8 B.x2 + 4x + 2 C.x3 + 3x2 – 2x + 8 D.x3 + 7x2 + 14x – 8 Example 6

  39. Section 1 (pg 307): • 17 – 61 odd, 62 & 67 (25 problems)

  40. End of the Lesson

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