EXAMPLE 1. a. Add 2x 3 5x 2 + 3x 9 and x 3 + 6x in a vertical format. SOLUTION. a. 2x 3 5x 2 + 3x 9 + x 3 + 6x x 3 + x 2 + 3x + 2
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1 EXAMPLE 1 Add polynomials vertically and horizontally a. Add 2x 3 5x 2 + 3x 9 and x 3 + 6x in a vertical format. a. 2x 3 5x 2 + 3x 9 + x 3 + 6x x 3 + x 2 + 3x + 2
2 EXAMPLE 1 Add polynomials vertically and horizontally b. Add 3y 3 2y 2 7y and 4y 2 + 2y 5 in a horizontal format. (3y 3 2y 2 7y) + ( 4y 2 + 2y 5) = 3y 3 2y 2 4y 2 7y + 2y 5 = 3y 3 6y 2 5y 5
3 EXAMPLE 2 Subtract polynomials vertically and horizontally a. Subtract 3x 3 + 2x 2 x + 7 from 8x 3 x 2 5x + 1 in a vertical format. a. Align like terms, then add the opposite of the subtracted polynomial. 8x 3 x 2 5x + 1 (3x 3 + 2x 2 x + 7) 8x 3 x 2 5x x 3 2x 2 + x 7 5x 3 3x 2 4x 6
4 EXAMPLE 2 Subtract polynomials vertically and horizontally b. Subtract 5z 2 z + 3 from 4z 2 + 9z 12 in a horizontal format. Write the opposite of the subtracted polynomial, then add like terms. (4z 2 + 9z 12) (5z 2 z + 3) = 4z 2 + 9z 12 5z 2 + z 3 = 4z 2 5z 2 + 9z + z 12 3 = z z 15
5 GUIDED PRACTICE for Examples 1 and 2 Find the sum or difference. 1. (t 2 6t + 2) + (5t 2 t 8) t 2 6t t 2 t 8 6t 2 7t 6
6 GUIDED PRACTICE for Examples 1 and 2 2. (8d 3 + 9d 3 ) (d 3 13d 2 4) = (8d 3 + 9d 3 ) (d 3 13d 2 4) = (8d 3 + 9d 3 ) d d 2 + 4) = 9d 3 3 d d 2 + 8d = 8d d 2 + 8d + 1
7 EXAMPLE 3 Multiply polynomials vertically and horizontally a. Multiply 2y 2 + 3y 6 and y 2 in a vertical format. b. Multiply x + 3 and 3x 2 2x + 4 in a horizontal format. a. 2y 2 + 3y 6 y 2 4y 2 6y + 12 Multiply 2y 2 + 3y 6 by 2. 2y 3 + 3y 2 6y Multiply 2y 2 + 3y 6 by y 2y 3 +7y 2 12y + 12 Combine like terms.
8 EXAMPLE 3 Multiply polynomials vertically and horizontally b. (x + 3)(3x 2 2x + 4) = (x + 3)3x 2 (x + 3)2x + (x + 3)4 = 3x 3 + 9x 2 2x 2 6x + 4x + 12 = 3x 3 + 7x 2 2x + 12
9 EXAMPLE 4 Multiply three binomials Multiply x 5, x + 1, and x + 3 in a horizontal format. (x 5)(x + 1)(x + 3) = (x 2 4x 5)(x + 3) = (x 2 4x 5)x + (x 2 4x 5)3 = x 3 4x 2 5x + 3x 2 12x 15 = x 3 x 2 17x 15
10 EXAMPLE 5 Use special product patterns a. (3t + 4)(3t 4) = (3t) Sum and difference = 9t 2 16 b. (8x 3) 2 = (8x) 2 2(8x)(3) Square of a binomial = 64x 2 48x + 9 c. (pq + 5) 3 Cube of a = (pq) 3 + 3(pq) 2 (5) + 3(pq)(5) = p 3 q p 2 q pq binomial
11 GUIDED PRACTICE for Examples 3, 4 and 5 Find the product. 3. (x + 2)(3x 2 x 5) 3x 2 x 5 x + 2 6x 2 2x 10 3x 3 x 2 5x 3x 3 + 5x 2 7x 10 Multiply 3x 2 x 5 by 2. Multiply 3x 2 x 5 by x. Combine like terms.
12 GUIDED PRACTICE for Examples 3, 4 and 5 4. (a 5)(a + 2)(a + 6) (a 5)(a + 2)(a + 6) = (a 2 3a 10)(a + 6) = (a 2 3a 10)a + (a 2 3a 10)6 = (a 3 3a 2 10a + 6a 2 18a 60) = (a 3 + 3a 2 28a 60)
13 GUIDED PRACTICE for Examples 3, 4 and 5 5. (xy 4) 3 (xy 4) 3 = (xy) 3 3(xy) 2 + 3(xy)(4) 2 (4) 3 = x 3 y 3 12x 2 y xy 64
14 EXAMPLE 6 Petroleum Use polynomial models 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 = 0.575t t and O = 0.249t where t is the number of years since Write a model for the average total amount T of crude oil produced per day. What was the average total amount of crude oil produced per day in 2000?
15 EXAMPLE 6 Use polynomial models To find a model for T, multiply the two given models t t t t t t t t t t t
16 EXAMPLE 6 Use polynomial models ANSWER Total daily oil output can be modeled by T = 0.143t t t 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.
17 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 Write a model for the average total cost T of drilling a new well. D = 109t C = 0.542t t
18 GUIDED PRACTICE for Example 6 To find a model for T, multiply the two given models t t t t t t t t t t t
19 GUIDED PRACTICE for Example 6 ANSWER Total daily oil output can be modeled by T = t t 2 20,057t
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