WarmUp Exercises Simplify the expression 1 3 x

Warm-Up Exercises Simplify the expression. 1. (– 3 x 3)(5 x) ANSWER – 15 x 4 2. 9 x – 18 x ANSWER – 9 x 3. 10 y 2 + 7 y – 8 y 2 – 1 ANSWER 2 y 2 + 7 y – 1

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

Warm-Up 1 Exercises EXAMPLE Add polynomials vertically and horizontally a. Add 2 x 3 – 5 x 2 + 3 x – 9 and x 3 + 6 x 2 + 11 in a vertical format. SOLUTION a. 2 x 3 – 5 x 2 + 3 x – 9 + x 3 + 6 x 2 + 11 3 x 3 + x 2 + 3 x + 2

Warm-Up 1 Exercises EXAMPLE Add polynomials vertically and horizontally b. Add 3 y 3 – 2 y 2 – 7 y and – 4 y 2 + 2 y – 5 in a horizontal format. (3 y 3 – 2 y 2 – 7 y) + (– 4 y 2 + 2 y – 5) = 3 y 3 – 2 y 2 – 4 y 2 – 7 y + 2 y – 5 = 3 y 3 – 6 y 2 – 5 y – 5

Warm-Up 2 Exercises EXAMPLE Subtract polynomials vertically and horizontally a. Subtract 3 x 3 + 2 x 2 – x + 7 from 8 x 3 – x 2 – 5 x + 1 in a vertical format. SOLUTION a. Align like terms, then add the opposite of the subtracted polynomial. 8 x 3 – x 2 – 5 x + 1 – (3 x 3 + 2 x 2 – x + 7) 8 x 3 – x 2 – 5 x + 1 + – 3 x 3 – 2 x 2 + x – 7 5 x 3 – 3 x 2 – 4 x – 6

Warm-Up 2 Exercises EXAMPLE Subtract polynomials vertically and horizontally b. Subtract 5 z 2 – z + 3 from 4 z 2 + 9 z – 12 in a horizontal format. Write the opposite of the subtracted polynomial, then add like terms. (4 z 2 + 9 z – 12) – (5 z 2 – z + 3) = 4 z 2 + 9 z – 12 – 5 z 2 + z – 3 = 4 z 2 – 5 z 2 + 9 z + z – 12 – 3 = – z 2 + 10 z – 15

Warm-Up YOU TRY. . Exercises. for Examples 1 and 2 Find the sum or difference. 1. (t 2 – 6 t + 2) + (5 t 2 – t – 8) SOLUTION t 2 – 6 t + 2 + 5 t 2 – t – 8 6 t 2 – 7 t – 6

Warm-Up YOU TRY. . Exercises. for Examples 1 and 2 2. (8 d – 3 + 9 d 3) – (d 3 – 13 d 2 – 4) SOLUTION = (8 d – 3 + 9 d 3) – (d 3 – 13 d 2 – 4) = (8 d – 3 + 9 d 3) – d 3 + 13 d 2 + 4) = 9 d 3 – 3 d 3 + 13 d 2 + 8 d – 3 + 4 = 8 d 3 + 13 d 2 + 8 d + 1

Warm-Up 3 Exercises EXAMPLE Multiply polynomials vertically and horizontally a. Multiply – 2 y 2 + 3 y – 6 and y – 2 in a vertical format. b. Multiply x + 3 and 3 x 2 – 2 x + 4 in a horizontal format. SOLUTION a. – 2 y 2 + 3 y – 6 y– 2 4 y 2 – 6 y + 12 Multiply – 2 y 2 + 3 y – 6 by – 2 y 3 + 3 y 2 – 6 y Multiply – 2 y 2 + 3 y – 6 by y – 2 y 3 +7 y 2 – 12 y + 12 Combine like terms.

Warm-Up 3 Exercises EXAMPLE Multiply polynomials vertically and horizontally b. (x + 3)(3 x 2 – 2 x + 4) = (x + 3)3 x 2 – (x + 3)2 x + (x + 3)4 = 3 x 3 + 9 x 2 – 2 x 2 – 6 x + 4 x + 12 = 3 x 3 + 7 x 2 – 2 x + 12

Warm-Up 4 Exercises EXAMPLE Multiply three binomials Multiply x – 5, x + 1, and x + 3 in a horizontal format. (x – 5)(x + 1)(x + 3) = (x 2 – 4 x – 5)x + (x 2 – 4 x – 5)3 = x 3 – 4 x 2 – 5 x + 3 x 2 – 12 x – 15 = x 3 – x 2 – 17 x – 15

Warm-Up Exercises

Warm-Up 5 Exercises EXAMPLE Use special product patterns a. (3 t + 4)(3 t – 4) = (3 t)2 – 42 Sum and difference = 9 t 2 – 16 b. (8 x – 3)2 = (8 x)2 – 2(8 x)(3) + 32 Square of a binomial = 64 x 2 – 48 x + 9 c. (pq + 5)3 = (pq)3 + 3(pq)2(5) + 3(pq)(5)2 + 53 Cube of a binomial = p 3 q 3 + 15 p 2 q 2 + 75 pq + 125

Warm-Up YOU TRY. Exercises. . for Examples 3, 4 and 5 Find the product. 3. (x + 2)(3 x 2 – x – 5) SOLUTION 3 x 2 – x – 5 x+2 6 x 2 – 2 x – 10 3 x 3 – x 2 – 5 x Multiply 3 x 2 – x – 5 by 2. 3 x 3 + 5 x 2 – 7 x – 10 Combine like terms. Multiply 3 x 2 – x – 5 by x.

Warm-Up YOU TRY. . Exercises. for Examples 3, 4 and 5 4. (a – 5)(a + 2)(a + 6) SOLUTION (a – 5)(a + 2)(a + 6) = (a 2 – 3 a – 10)a + (a 2 – 3 a – 10)6 = (a 3 – 3 a 2 – 10 a + 6 a 2 – 18 a – 60) = (a 3 + 3 a 2 – 28 a – 60)

Warm-Up YOU TRY. . Exercises. for Examples 3, 4 and 5 5. (xy – 4)3 SOLUTION (xy – 4)3 = (xy)3 – 3(xy)2 + 3(xy)(4)2 – (4)3 = x 3 y 3 – 12 x 2 y 2 + 48 xy – 64

Warm-Up 6 Exercises 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. 575 t 2 + 10. 9 t + 548 and O = – 0. 249 t + 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. What was the average total amount of crude oil produced per day in 2000?

Warm-Up 6 Exercises EXAMPLE Use polynomial models SOLUTION To find a model for T, multiply the two given models. – 0. 575 t 2 + – – 10. 9 t + 0. 249 t + 548 15. 4 8. 855 t 2 + 167. 86 t + 8439. 2 0. 143175 t 3 – 2. 7141 t 2 – 136. 452 t 0. 143175 t 3 – 11. 5691 t 2 + 31. 408 t + 8439. 2

Warm-Up 6 Exercises EXAMPLE Use polynomial models ANSWER Total daily oil output can be modeled by T = 0. 143 t 3 – 11. 6 t 2 + 31. 4 t + 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 YOU TRY. . Exercises. 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 = 109 t + 4010 C = 0. 542 t 2 – 7. 16 t + 79. 4

Warm-Up YOU TRY. . Exercises. for Example 6 SOLUTION To find a model for T, multiply the two given models. 0. 542 t 2 – 7. 16 t + 79. 4 109 t + 4010 2173. 68 t 2 + 28711. 6 t + 318394 59. 078 t 3 – 780. 44 t 2 – 8654. 6 t 59. 078 t 3 + 1392. 98 t 2 – 20057 t + 318394

Warm-Up YOU TRY. . Exercises. ANSWER for Example 6 Total daily oil output can be modeled by T = 59. 078 t 3 + 1392. 98 t 2 – 20, 057 t + 318394.

Warm-Up Exercises KEEP GOING Find the sum, difference, or product. 1. (3 x 2 + 5 x + 2) + (x 2 – 3 x + 6) ANSWER 4 x 2 + 2 x + 8 2. (5 p 3 + 2 p 2 – 3 p – 7) - (2 p 3 – 4 p 2 – 5 p + 6) ANSWER (3 p 3 + 6 p 2 + 2 p – 13)

Warm-Up Exercises KEEP GOING 3. (5 a 2 + 6 a + 9)(2 a – 3) ANSWER 10 a 3 -3 a^2 -18 4. 6(x – 1)(x + 1) ANSWER 6 x 2 – 6

Warm-Up Exercises KEEP GOING 5. The floor space in square feet of retail stores A, B and C can be modeled by A = x 2 + 4 x – 7, B = 2 x 2 – 7 x + 1 and C = – 4 x 2 + 250 x – 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 – x 2 + 247 x – 7; 9843 ft 2.