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

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.

EXAMPLE 1 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

EXAMPLE 1 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

EXAMPLE 2 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 3x3 + 2x2 – x + 7 8x3 – x2 – 5x + 1 + – 3x3 – 2x2 + x – 7 5x3 – 3x2 – 4x – 6

EXAMPLE 2 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

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

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

Multiply polynomials vertically and horizontally EXAMPLE 3 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 a. – 2y2 + 3y – 6 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.

EXAMPLE 3 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

EXAMPLE 4 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

Use special product patterns EXAMPLE 5 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 Square of a binomial = 64x2 – 48x + 9

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

Expanding a binomial using Pascal’s Triangle Pascal’s Triangle 1 1 2 1 1 2 1 1 3 3 1 1 4 6 4 1 Write the next row. 1 5 10 10 5 1 1 6 15 20 15 6 1 Cube of a binomial Expand (pq + 5)3 = (pq)3 + 3(pq)2(5) + 3(pq)(5)2 + 53 = p3q3 + 15p2q2 + 75pq + 125

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.

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 Multiply 3x2 – x – 5 by 2 . 3x3 – x2 – 5x Multiply 3x2 – x – 5 by x . 3x3 + 5x2 – 7x – 10 Combine like terms.

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)

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

EXAMPLE 6 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?

EXAMPLE 6 Use polynomial models SOLUTION To find a model for T, multiply the two given models. – 0.575t2 + 10.9t + 548 – 0.249t + 15.4 – 8.855t2 + 167.86t + 8439.2 0.143175t3 – 2.7141t2 – 136.452t 0.143175t3 – 11.5691t2 + 31.408t + 8439.2

EXAMPLE 6 Use polynomial models 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. ANSWER

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

GUIDED PRACTICE for Example 6 SOLUTION To find a model for T, multiply the two given models. 2173.68t2 + 28711.6t + 318394 0.542t2 – 7.16t + 79.4 109t + 4010 59.078t3 – 780.44t2 – 8654.6t 59.078t3 + 1392.98t2 – 20057t + 318394

GUIDED PRACTICE for Example 6 Total daily oil output can be modeled by T = 59.078t3 + 1392.98t2 – 20,057t + 318394. ANSWER

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)

Daily Homework Quiz 3. (5a2 + 6a + 9)(2a – 3) ANSWER 8a3 – 27 4. 6(x – 1)(x + 1) ANSWER 6x2 – 6

Daily Homework Quiz 5. The floor space in square feet of retail stores A, B and C can be modeled by and 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? A = x2 + 4x – 7, B = 2x2 – 7x + 1 C = – 4x2 + 250x – 1. ANSWER – x2 + 247x – 7; 9843 ft2.