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Exploring Polynomials

This lesson explores polynomials by solving a problem of determining the number of telephone lines needed for a set of buildings and practicing skills of simplifying polynomial expressions.

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Exploring Polynomials

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  1. Exploring Polynomials LESSON 12-1 Problem of the Day The telephone company is installing telephone lines for ten buildings. Each building is to be connected to each of the other buildings with one line. How many telephone lines are needed? 45 12-1

  2. 1 2 Exploring Polynomials LESSON 12-1 Check Skills You’ll Need (For help, go to Lesson 6-2.) • Vocabulary Review Are x, x, and – 4x like terms? • Simplify each expression. • 2. –2 + 2t – 3t • 3. 7w – 10 + 5w • 4. 3k + 32k – 5 Check Skills You’ll Need 12-1

  3. Exploring Polynomials LESSON 12-1 Check Skills You’ll Need Solutions 1. yes; all the terms include the same variable, x 2. –2 + 2t – 3t = –2 + (2t – 3t) = –2 + (2 – 3)t = –2 – t 3. 7w – 10 + 5w = 7w + 5w – 10 = (7w + 5w) – 10 = (7 + 5)w – 10 = 12w – 10 4. 3k + 32k – 5 = (3k + 32k) – 5 = (3+ 32)k – 5 = 35k – 5 12-1

  4. Exploring Polynomials LESSON 12-1 Additional Examples Write a variable expression for this model. The model shows the expression x2 – 2x + 3. Quick Check 12-1

  5. Model each term. Group like terms together. Remove zero pairs. Exploring Polynomials LESSON 12-1 Additional Examples Use tiles to simplify the polynomial –x2 + 3x + x2 + x2 + 3 – x – 4. Quick Check The simplified polynomial is x2 + 2x – 1. 12-1

  6. Use the Commutative Property. = 4x2 + 2x2 +2x – 5x + 3 – 5 Use the Associative Property. = (4x2 + 2x2 ) +(2x – 5x) + (3 – 5) = (4 + 2)x2 +(2– 5)x + (3 – 5) Use the Distributive Property. Exploring Polynomials LESSON 12-1 Additional Examples A playground has areas of grass and sand. The polynomial 4x2 – 5 + 2x + 2x2 + 3 – 5x represents the total area of the grass minus the sandy areas. Use properties of numbers to simplify the polynomial. 4x2 – 5 + 2x + 2x2 + 3 – 5x = 6x2 – 3x – 2 The total area of the grass can be represented by 6x2 – 3x – 2. Quick Check 12-1

  7. Exploring Polynomials LESSON 12-1 Lesson Quiz 1. Write a variable expression for the model. 2. Write a variable expression for the model. x2 – 2x + 4 –x2 + 3x – 1 3. Use properties to simplify the polynomial 4x2 + 9x – x2 – 10x – 2. 3x2 – x – 2 4. Use properties to simplify the polynomial 5a2 – 7a + 4 + a – 7a2 + 6. –2a2 – 6a + 10 12-1

  8. Adding and Subtracting Polynomials LESSON 12-2 Problem of the Day Bianca’s family needs to choose the exterior paint for their new house. The wall colors are white, green, and beige. The trim colors are white, green, blue, and cocoa. How many combinations of wall color and trim are possible? 12 12-2

  9. Adding and Subtracting Polynomials LESSON 12-2 Check Skills You’ll Need (For help, go to Lesson 12-1.) • 1. Vocabulary Review Name the constant in the • polynomial 1 – p + 2p. • Simplify. • 2. 2y2 + 3y + (–5y) • 3. –x + x + 6x2 – 1 • 4. 7z – 8z2 + z + 3z2 Check Skills You’ll Need 12-2

  10. Adding and Subtracting Polynomials LESSON 12-2 Check Skills You’ll Need Solutions 1. 1 2. 2y2 + 3y + (–5y) = 2y2 + [3 + (–5)]y = 2y2 – 2y 3. –x + x + 6x2 – 1 = (–1 + 1)x + 6x2 – 1 = 6x2 – 1 4. 7z – 8z2 + z + 3z2 = (7z + z) + (–8z2 + 3z2) = 8z – 5z2 12-2

  11. Method 1 Add using tiles. Adding and Subtracting Polynomials LESSON 12-2 Additional Examples Add: (5p2 + 2p + 7) + (2p2 – p – 5). 12-2

  12. (5p2 + 2p + 7) + (2p2 – p – 5) = (5p2 + 2p2) + (2p – p) + (7 – 5) Group like terms. = 7p2 + p + 2 Simplify. = (5 + 2)p2 + (2 – 1)p + (7 – 5) Use the Distributive Property. Check Check the solution in Example 1 by substituting 1 for p. (5p2 + 2p + 7) + (2p2 – p – 5) 7p2 + p + 2 (5 • 12 + 2 • 1 + 7) + (2 • 12 – 1 – 5) (7 • 12 + 1 + 2) (5 + 2 + 7) + (2 – 1 – 5) (7 + 1 + 2) 10 = 10 Substitute 1 for p. Multiply. Add. Adding and Subtracting Polynomials LESSON 12-2 Additional Examples Quick Check (continued) Method 2 Add using properties. 12-2

  13. = (3x + 4x + 5x + 7x) + (5 – 2 + 2 – 6) Group like terms. Add the coefficients. = 19x – 1 Adding and Subtracting Polynomials LESSON 12-2 Additional Examples Quick Check A garden has sides of 3x + 5, 4x – 2, 5x + 2, and 7x – 6. Write a polynomial to express the length of edging that is needed to go around the garden. To find the perimeter of the garden, find the sum of the four sides. perimeter = (3x + 5) + (4x – 2) + (5x + 2) + (7x – 6) The perimeter of the garden is (19x – 1). The edging must be (19x – 1) units long to go around the garden. 12-2

  14. Add the opposite of each term in the second polynomial. = (3q2 – 2q + 4) + (–2q2 + 2q – 3) = (3q2 – 2q2) + (–2q + 2q) + (4 – 3) Group like terms. = (3 – 2)q2 + (–2 + 2)q + (4 – 3) Use the Distributive Property. = q2 + 1 Simplify. Adding and Subtracting Polynomials LESSON 12-2 Additional Examples Subtract: (3q2 – 2q + 4) – (2q2 – 2q + 3). (3q2 – 2q + 4) – (2q2 – 2q + 3) Quick Check 12-2

  15. Adding and Subtracting Polynomials LESSON 12-2 Lesson Quiz Simplify each polynomial. 1. (4n2 + n + 1) + (n2 + 3n + 1) 2. (x2 – 2x + 6) + (x2 + 2x – 2) 3. (a2 – 7) – (a2 + 4a – 4) 4. (m – 5) + (m2 – 12) + (6m2 – 9m) 5n2 + 4n + 2 2x2 + 4 –4a – 3 7m2– 8m – 17 12-2

  16. 1 2 24 ft2 Exponents and Multiplication LESSON 12-3 Problem of the Day 1 2 Find the area of a rectangle 3 ft wide and twice as high. 12-3

  17. Exponents and Multiplication LESSON 12-3 Check Skills You’ll Need (For help, go to Lesson 2-7.) • Vocabulary Review What is the base of the exponential expression xy? • Simplify each expression. • 2. (–1)4 • 3. (–3)2 • 4. –32 • 5. –14 Check Skills You’ll Need 12-3

  18. Exponents and Multiplication LESSON 12-3 Check Skills You’ll Need Solutions 1. x 2. (–1)4 = (–1) • (–1) • (–1) • (–1) = 1 3. (–3)2 = (–3) • (–3) = 9 4. –32 = –1(3 • 3) = –9 5. –14 = –1(1 • 1 • 1 • 1) = –1 12-3

  19. (–3)2 (–3)4 Add the exponents. (–3)2 (–3)4 =(–3)(2+4) Simplify the exponent. = (–3)(6) Exponents and Multiplication LESSON 12-3 Additional Examples Write the expression using a single exponent. Quick Check 12-3

  20. Use the Associative and Commutative properties. (3103) (7105) = (3 • 7)  (103 • 105) = 21 (103 • 105) Multiply 3 and 7. Add the exponents for the powers of 10. = 21108 Write 21 in scientific notation. = 2.1  101 108 = 2.1  109 Add the exponents. Exponents and Multiplication LESSON 12-3 Additional Examples Quick Check Multiply (3 x 103) (7 x 105). Write the product in scientific notation. 12-3

  21. (5.9 x 1012) (1.609 x 103) Multiply by the conversion factor. = (5.9 1.609) x (1012 103) Use the Associative and Commutative properties. Multiply 5.9 and 1.609. Round to the nearest tenth. 9.5 x (1012 103) = 9.5 x 1015 Add exponents of the powers of 10. Exponents and Multiplication LESSON 12-3 Additional Examples A light-year is 5.9 x 1012 miles. A mile is 1.609 x 103 meters. In scientific notation, about how many meters are in a light year? Quick Check 12-3

  22. Exponents and Multiplication LESSON 12-3 Lesson Quiz 1. Write (–8)4 (–8)5 using a single exponent. 2. Write the product of (8.2 x 106) and (5 x 102) in scientific notation. 3. The speed of light is 3.00 x 105 km/s. Find the distance light travels in 8 x 102 seconds. 4. A light-year is 5.9 x 1012 miles. A mile is approximately 6.34 x 104 inches. About how many inches are in a light-year? • (–8)9 4.1 x 109 2.4 x 108 km 3. 74 x 1017 in 12-3

  23. Multiplying Polynomials LESSON 12-4 Problem of the Day Palindromes are numbers, words, or sentences that read the same forward and backward. Find a number to add to each number to get a palindrome as a sum. 175.3 60.32 0.271 1.94 12-4

  24. Multiplying Polynomials LESSON 12-4 Check Skills You’ll Need (For help, go to Lesson 12-3.) 1. Vocabulary Review The expression 23 • 25 can be simplified by adding the Simplify using a single exponent. 2. x4 • x5 • x6 3. (–a)3 • (–a)7 Check Skills You’ll Need 12-4

  25. Multiplying Polynomials LESSON 12-4 Check Skills You’ll Need Solutions 1. exponents 2 . x4 • x5 • x6 = x (4+ 5 + 6) = x15 3 . (–a)3 • (–a)7 = (–a)(3+7) = (–a)10 = a10 12-4

  26. Use the Commutative Property to rearrange the factors. (3x2)(–4x3) = (3)(–4) • x2 • x3 Multiply the coefficients. = –12 • x2 • x3 = –12x5 Add the exponents. Multiplying Polynomials LESSON 12-4 Additional Examples Simplify (3x2)(–4x3). Quick Check 12-4

  27. A = l w = 2x(4x + 6) = 2x 4x + 2x 6 Multiplying Polynomials LESSON 12-4 Additional Examples If the width of a new house foundation is represented by 2x and the length is represented by 4x + 6, which expression represents the area of the foundation? To find the area, multiply the width, 2x, times the length, 4x + 6. = 8x2 + 12x Quick Check 12-4

  28. Count each type of tile. There are three x2 tiles. There are four x tiles. There is one unit tile. Multiplying Polynomials LESSON 12-4 Additional Examples Use an area model to simplify (x + 1)(3x + 1). So, (x + 1)(3x + 1) = 3x2 + 4x + 1. Quick Check 12-4

  29. Multiplying Polynomials LESSON 12-4 Lesson Quiz Simplify each expression in 1–3. 1. (–7c3)(4c) 2. x(3x + 2) 3. –2m2(m – 1) –28c4 3x2 + 2x –2m3 + 2m2 4. If the base of a parallelogram-shaped playground is represented by 6y, and the height is represented by 4y – 3, which expression represents the area? 24y2 – 18y 12-4

  30. 1 2 1 2 3 4 1 yd2 Exponents and Division LESSON 12-5 Problem of the Day Lupe’s flag is a square with 1 yd on a side. Anna makes her flag yd larger on each side. How much larger is Anna’s flag than Lupe’s? 12-5

  31. Exponents and Division LESSON 12-5 Check Skills You’ll Need (For help, go to Lesson 2-7.) 1. Vocabulary Review Is the expression x5 an exponent or a power? Write each expression using exponents. 2. 7 • 7 • 7 • 7 3. 4 • 4 • 4 4. 5 • 5 5. 1 • 1 • 1 • 1 • 1 Check Skills You’ll Need 12-5

  32. Exponents and Division LESSON 12-5 Check Skills You’ll Need Solutions 1. power 2. 7 is a factor 4 times: 74 3. 4 is a factor 3 times: 43 4. 5 is a factor 2 times: 52 5. 1 is a factor 5 times: 15 12-5

  33. = x(14 – 9) Subtract exponents with the same base. x14 x9 x14 x9 = x5 Simplify. Exponents and Division LESSON 12-5 Additional Examples Write using a single exponent. Quick Check 12-5

  34. distance speed distance speed Use the formula for time. time = 4.84 1.1 108 107 Substitute. Write as a product of quotients. x = 101 4.84 1.1 Subtract exponents. x = Divide. 101 x 4.4 Exponents and Division LESSON 12-5 Additional Examples Quick Check The distance between the sun and Jupiter is about 4.84 x 108 miles. Light travels at about 1.1 x 107 miles per minute. Use the formula time = to estimate how long it takes sunlight to reach Jupiter. Write your answer in standard form. It takes about 4.4 x 101 minutes, or 44 minutes, for sunlight to reach Jupiter. 12-5

  35. (–5)0 = 1 Simplify. Simplify. 5y0 = 5  1 = 5 Exponents and Division LESSON 12-5 Additional Examples Simplify each expression. a. (–5)0 b. 5y0 Quick Check 12-5

  36. 1 p8 1 23 1 8 Write the expression using a positive exponent. 2–3 = = Simplify. Exponents and Division LESSON 12-5 Additional Examples Simplify each expression. a. 2–3 b. (p) –8 Quick Check 12-5

  37. 1 16 97 94 Exponents and Division LESSON 12-5 Lesson Quiz 1. Write using a single exponent. Simplify each expression. 2. 60 3. 4–2 4. Jupiter’s diameter is about 1.43 x 105 kilometers. Earth’s diameter is about 1.28 x 104 kilometers. How many times greater is Jupiter’s diameter than Earth’s diameter? Round the answer to the nearest tenth. 93 1 11.2 times greater 12-5

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