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Warm Up

Add, Subtract, and Multiply Polynomials. Warm Up. Lesson Presentation. Lesson Quiz. Warm-Up. Simplify the expression. 1. ( – 3 x 3 )(5 x ). – 15 x 4. ANSWER. 2. 9 x – 18 x. – 9 x. ANSWER. 3. 10 y 2 + 7 y – 8 y 2 – 1. 2y 2 + 7 y – 1. ANSWER. Warm-Up.

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Warm Up

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  1. Add, Subtract, and Multiply Polynomials Warm Up Lesson Presentation Lesson Quiz

  2. Warm-Up Simplify the expression. 1. (–3x3)(5x) –15x4 ANSWER 2. 9x– 18x –9x ANSWER 3. 10y2 + 7y – 8y2 – 1 2y2 + 7y – 1 ANSWER

  3. Warm-Up Simplify the expression. 4. 4(–5a + 6) – 2(a – 8) –22a +40 ANSWER 5. Each side of a square is (2x + 5) inches long. Write an expression for the perimeter of the square. (8x + 20) in. ANSWER

  4. a.2x3 – 5x2 + 3x – 9 + x3 + 6x2 + 11 3x3 + x2 + 3x + 2 Example 1 a. Add 2x3 – 5x2 + 3x – 9 andx3 + 6x2 + 11 in a vertical format. SOLUTION

  5. Example 1 b. Add 3y3 – 2y2 – 7y and –4y2 + 2y – 5 in a horizontal format. (3y3 – 2y2 – 7y) + (–4y2 + 2y – 5) = 3y3 – 2y2 – 4y2 – 7y + 2y – 5 = 3y3 – 6y2 – 5y – 5

  6. 8x3 – x2 – 5x + 1 8x3 – x2 – 5x + 1 – (3x3 + 2x2–x + 7) + – 3x3– 2x2 + x – 7 5x3 – 3x2 – 4x – 6 Example 2 a. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format. SOLUTION a. Align like terms, then add the opposite of the subtracted polynomial.

  7. Example 2 b. Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal format. Write the opposite of the subtracted polynomial, then add like terms. (4z2 + 9z – 12) – (5z2–z + 3) = 4z2 + 9z – 12 – 5z2 + z – 3 = 4z2 – 5z2 + 9z + z – 12 – 3 = –z2 + 10z – 15

  8. Guided Practice Find the sum or difference. 1. (t2 – 6t + 2) + (5t2 – t – 8) ANSWER 6t2 – 7t – 6 2. (8d – 3 + 9d3) – (d3 – 13d2 – 4) ANSWER 8d3 + 13d2 + 8d + 1

  9. a. –2y2+ 3y – 6 y – 2 4y2 – 6y + 12 –2y3 + 7y2 –12y + 12 Example 3 a. Multiply –2y2 + 3y – 6 andy – 2 in a vertical format. SOLUTION Multiply –2y2 + 3y – 6 by –2 . –2y3 + 3y2 – 6y Multiply –2y2 + 3y – 6 by y Combine like terms.

  10. Example 3 b. Multiplyx + 3 and 3x2 – 2x + 4 in a horizontal format. SOLUTION b.(x + 3)(3x2 – 2x + 4) = (x + 3)3x2 –(x + 3)2x +(x + 3)4 = 3x3 + 9x2 – 2x2 – 6x + 4x + 12 = 3x3 + 7x2 – 2x + 12

  11. Example 4 Multiply x – 5, x + 1, and x + 3 in a horizontal format. (x – 5)(x + 1)(x+ 3)= (x2– 4x – 5)(x + 3) = (x2– 4x – 5)x +(x2– 4x – 5)3 = x3 – 4x2 – 5x + 3x2 – 12x – 15 = x3 – x2 – 17x – 15

  12. Example 5 a. (3t + 4)(3t – 4) = (3t)2 – 42 Sum and difference = 9t2 – 16 b. (8x – 3)2 = (8x)2 – 2(8x)(3) + 32 Square of a binomial = 64x2 – 48x + 9 Cube of abinomial c. (pq + 5)3 = (pq)3 + 3(pq)2(5) + 3(pq)(5)2 + 53 = p3q3 + 15p2q2 + 75pq + 125

  13. Guided Practice Find the product. 3. (x + 2)(3x2 – x – 5) ANSWER 3x3 + 5x2 – 7x – 10 4. (a – 5)(a + 2)(a + 6) (a3 + 3a2 – 28a – 60) ANSWER 5. (xy – 4)3 x3y3 – 12x2y2 + 48xy – 64 ANSWER

  14. Example 6 Petroleum From 1986 to 2007, the number W(in thousands) of U.S. wells producing crude oil and the average daily oil output per well O(in barrels) can be modeled by the functions W = –6.49t + 631 andO = 0.00764t2–0.310t + 13.7 where tis the number of years since 1986. Write a model for the average total amount Tof crude oil produced per day. What was the average total amount of crude oil produced per day in 2008?

  15. Example 6 SOLUTION To find a model for T, multiply the two given models. 0.00764t2 – 0.310t + 13.7 – 6.49t + 631 4.8208t2 + 195.61t + 8644.7 – 0.049584t3 + 2.0119t2 – 88.913t – 0.049584t3 + 6.8327t2 – 284.523t + 8644.7 ANSWER Total daily oil output can be modeled byT = –0.049584t3 + 6.8327t2 – 284.523t + 8644.7 where Tis measured in thousands of barrels. By substitutingt = 22 into the model, you can estimate that the average total amount of crude oil produced per day in 2008 was about 5150 thousand barrels, or 4,930,000 barrels.

  16. Total daily oil output can be modeled by T = 59.078t3 + 1392.98t2 – 20,057t + 318,394. ANSWER 6. From 1994 to 2006, the number W (in thousands) of U.S. wells producing natural gas and the average daily gas output per well O (in cubic feet) can be modeled by the functions below where t is the number of years since 1994. Write a model for the average total amount T of natural gas produced daily. Guided Practice Industry W = 12.2t + 275 O = –0.181t2 – 2.06t + 164

  17. 1. (3x2 + 5x + 2) + (x2 – 3x + 6) ANSWER 4x2 + 2x + 8 2. (5p3 + 2p2 – 3p – 7) – (2p3 – 4p2 – 5p + 6) ANSWER (3p3 + 6p2 + 2p – 13) Lesson Quiz Find the sum, difference, or product.

  18. 3. (4a2 + 6a + 9)(2a – 3) ANSWER 8a3 – 27 4. 6(x – 1)(x + 1) ANSWER 6x2 – 6 Lesson Quiz Find the sum, difference, or product.

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