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Warm-Up Exercises Simplify the expression. 1. (–3x3)(5x) ANSWER –15x4 2. 9x – 18x ANSWER –9x 3. 10y2 + 7y – 8y2 – 1 ANSWER 2y2 + 7y – 1 Warm-Up Exercises Simplify the expression. 4. 4(– 5a + 6) –2(a – 8) ANSWER –4 5. Each side of a square is (2x + 5) inches long. Write an expression for the perimeter of the square. ANSWER (8x + 20) in. Warm-Up1Exercises EXAMPLE Add polynomials vertically and horizontally a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x2 + 11 in a vertical format. SOLUTION a. 2x3 – 5x2 + 3x – 9 + x3 + 6x2 + 11 3x3 + x2 + 3x + 2 Warm-Up1Exercises EXAMPLE Add polynomials vertically and horizontally 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 Warm-Up2Exercises EXAMPLE Subtract polynomials vertically and horizontally a. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format. SOLUTION a. Align like terms, then draw your line and change your signs (of the bottom row and add straight down). 8x3 – x2 – 5x + 1 8x3 – x2 – 5x + 1 3x3 + 2x2 – x + 7 + – 3x3 – 2x2 + x – 7 5x3 – 3x2 – 4x – 6 Warm-Up2Exercises EXAMPLE Subtract polynomials vertically and horizontally b. Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal format. Distribute the negative sign to ALL 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 Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Find the sum or difference. 1. (t2 – 6t + 2) + (5t2 – t – 8) SOLUTION t2 – 6t + 2 + 5t2 – t – 8 6t2 – 7t – 6 Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 2. (8d – 3 + 9d 3) – (d 3 – 13d 2 – 4) SOLUTION = (8d – 3 + 9d 3) – (d 3 – 13d 2 – 4) = (8d – 3 + 9d 3) – d 3 + 13d 2 + 4 = 9d 3 –3 d 3 + 13d 2 + 8d – 3 + 4 = 8d 3 + 13d 2 + 8d + 1 Warm-Up3Exercises EXAMPLE Multiply polynomials vertically and horizontally a. Multiply – 2y2 + 3y – 6 and y – 2 in a vertical format. b. Multiply x + 3 and 3x2 – 2x + 4 in a horizontal format. SOLUTION – 2y2 + 3y – 6 a. y–2 4y2 – 6y + 12 Multiply – 2y2 + 3y – 6 by – 2 . – 2y3 + 3y2 – 6y Multiply – 2y2 + 3y – 6 by y – 2y3 +7y2 –12y + 12 Combine like terms. Warm-Up3Exercises EXAMPLE Multiply polynomials vertically and horizontally 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 Warm-Up4Exercises EXAMPLE Multiply three binomials 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 Warm-Up5Exercises EXAMPLE Use special product patterns a. (3t + 4)(3t – 4) = (3t)2 – 42 Sum and difference = 9t2 – 16 b. (8x – 3)2 = (8x)2 – 2(8x)(3) + 32 = 64x2 – 48x + 9 Square of a binomial Warm-Up Exercises The Binomial Theorem Let’s look at the expansion of (x + y)n (x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x2 +2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 Warm-Up Exercises Expanding a binomial using Pascal’s Triangle Pascal’s Triangle 1 1 1 1 2 1 1 1 1 1 3 4 5 6 3 6 10 15 1 4 10 20 1 5 15 Write the next row. 1 6 1 Expand (pq + 5)3 = (pq)3 + 3(pq)2(5) + 3(pq)(5)2 + 53 Cube of a binomial = p3q3 + 15p2q2 + 75pq + 125 Warm-Up Exercises From Pascal’s triangle write down the 4th row. Expand (x + 3)4 1 4 6 4 1 These numbers are the same numbers that are the coefficients of the binomial expansion. The expansion of (a + b)4 is: 1a4b0 + 4a3b1 + 6a2b2 + 4a1b3 + 1a0b4 Notice that the exponents always add up to 4 with the a’s going in descending order and the b’s in ascending order. Now substitute x in for a and 3 in for b. Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and 5 Find the product. 3. (x + 2)(3x2 – x – 5) SOLUTION 3x2 – x – 5 x+2 6x2 – 2x – 10 3x3 – x2 – 5x 3x3 + 5x2 – 7x – 10 Multiply 3x2 – x – 5 by 2 . Multiply 3x2 – x – 5 by x . Combine like terms. Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and 5 4. (a – 5)(a + 2)(a + 6) SOLUTION (a – 5)(a + 2)(a + 6) = (a2 – 3a – 10)(a + 6) = (a2 – 3a – 10)a + (a2 – 3a – 10)6 = (a3 – 3a2 – 10a + 6a2 – 18a – 60) = (a3 + 3a2 – 28a – 60) Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4 and 5 5. (xy – 4)3 SOLUTION (xy – 4)3 = (xy)3 + 3(xy)2(– 4) + 3(xy)(– 4)2 + (– 4)3 = x3y3 – 12x2y2 + 48xy – 64 Warm-Up6Exercises EXAMPLE Use polynomial models Petroleum Since 1980, the number W (in thousands) of United States wells producing crude oil and the average daily oil output per well O (in barrels) can be modeled by W = – 0.575t2 + 10.9t + 548 and O = – 0.249t + 15.4 where t is the number of years since 1980. Write a model for the average total amount T of crude oil produced per day (product of W and T). What was the average total amount of crude oil produced per day in 2000? Warm-Up6Exercises EXAMPLE Use polynomial models SOLUTION To find a model for T, multiply the two given models. – 0.575t2 + 10.9t + – 0.249t + 548 15.4 – 8.855t2 + 167.86t + 8439.2 0.143175t3 – 2.7141t2 – 136.452t 0.143175t3 – 11.5691t2 + 31.408t + 8439.2 Warm-Up6Exercises EXAMPLE Use polynomial models ANSWER Total daily oil output can be modeled by T = 0.143t3 – 11.6t2 + 31.4t + 8440 where T is measured in thousands of barrels. By substituting t = 20 into the model, you can estimate that the average total amount of crude oil produced per day in 2000 was about 5570 thousand barrels, or 5,570,000 barrels. Warm-Up Exercises GUIDED PRACTICE for Example 6 Industry 6. The models below give the average depth D (in feet) of new wells drilled and the average cost per foot C (in dollars) of drilling a new well. In both models, t represents the number of years since 1980. Write a model for the average total cost T of drilling a new well. D = 109t + 4010 C = 0.542t2 – 7.16t + 79.4 Warm-Up Exercises GUIDED PRACTICE for Example 6 SOLUTION To find a model for T, multiply the two given models. 0.542t2 – 7.16t + 79.4 109t + 4010 2173.68t2 + 28711.6t + 318394 59.078t3 – 780.44t2 – 8654.6t 59.078t3 + 1392.98t2 – 20057t + 318394 Warm-Up Exercises GUIDED PRACTICE ANSWER for Example 6 Total daily oil output can be modeled by T = 59.078t3 + 1392.98t2 – 20,057t + 318394. Warm-Up Exercises Daily Homework Quiz Find the sum, difference, or product. 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) Warm-Up Exercises Daily Homework Quiz 3. (5a2 + 6a + 9)(2a – 3) ANSWER 8a3 – 27 4. 6(x – 1)(x + 1) ANSWER 6x2 – 6 Warm-Up Exercises Daily Homework Quiz 5. The floor space in square feet of retail stores A, B and C can be modeled by A = x2 + 4x – 7, B = 2x2 – 7x + 1 and C = – 4x2 + 250x – 1. Write a model for the total Amount of floor space T for all three stores. What Is the total floor space of the three stores if x = 50? ANSWER – x2 + 247x – 7; 9843 ft2.