Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Warm-Up Exercises 1. Use the quadratic formula to solve 2x2 – 3x – 1 = 0. Round the nearest hundredth. ANSWER 1.78, –0.28 2. Use synthetic substitution to evaluate f (x) = x3 + x2 – 3x – 10 when x = 2. ANSWER –4 Warm-Up Exercises 3. A company’s income is modeled by the function P = 22x2 – 571x. What is the value of P when x = 200? ANSWER 765,800 Warm-Up1Exercises EXAMPLE Use polynomial long division Divide f (x) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5. SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient. Warm-Up1Exercises EXAMPLE Use polynomial long division 3x2 + 4x – 3 x2 – 3x + 5 )3x4 – 5x3 + 0x2 + 4x – 6 quotient Multiply divisor by 3x4/x2 = 3x2 3x4 – 9x3 + 15x2 4x3 – 15x2 + 4x Subtract. Bring down next term. 4x3 – 12x2 + 20x Multiply divisor by 4x3/x2 = 4x – 3x2 – 16x – 6 –3x2 + 9x – 15 – 25x + 9 Subtract. Bring down next term. Multiply divisor by – 3x2/x2 = – 3 remainder Warm-Up1Exercises EXAMPLE Use polynomial long division ANSWER 3x4 – 5x3 + 4x – 6 = 3x2 + 4x – 3 + – 25x + 9 x2 – 3x + 5 x2 – 3x + 5 CHECK You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend. (3x2 + 4x – 3)(x2 – 3x + 5) + (– 25x + 9) = 3x2(x2 – 3x + 5) + 4x(x2 – 3x + 5) – 3(x2 – 3x + 5) – 25x + 9 = 3x4 – 9x3 + 15x2 + 4x3 – 12x2 + 20x – 3x2 + 9x – 15 – 25x + 9 = 3x4 – 5x3 + 4x – 6 Warm-Up2Exercises EXAMPLE Use polynomial long division with a linear divisor Divide f (x) = x3 + 5x2 – 7x + 2 by x – 2. x2 + 7x + 7 x – 2 ) x3 + 5x2 – 7x + 2 x3 – 2x2 Multiply divisor by x3/x = x2. 7x2 – 7x Subtract. 7x2 – 14x 7x + 2 Multiply divisor by 7x2/x = 7x. 7x – 14 16 ANSWER quotient Subtract. Multiply divisor by 7x/x = 7. remainder x3 + 5x2 – 7x +2 = x2 + 7x + 7 + 16 x–2 x–2 Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Divide using polynomial long division. 1. (2x4 + x3 + x – 1) (x2 + 2x – 1) SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 2x2 – 3x + 8 x2 + 2x – 1 )2x4 + x3 + 0x2 + x – 1 Multiply divisor by 2x4/x2 = –2x2. 2x4 – 4x3 – 2x2 3x3 – 2x2 + x – 3x3 – 6x2 quotient + 3x Subtract. Bring down next term. Multiply divisor by –3x3/x2 = –3. 8x2 – 2x – 1 Subtract. Bring down next term. 8x2 –16x – 8 Multiply divisor by 4x2/x2 = 8. – 18x + 7 remainder Warm-Up Exercises GUIDED PRACTICE ANSWER for Examples 1 and 2 2x4 + 5x3 + x – 1 x2 + 2x – 1 = (2x2 – 3x + 8)+ –218x + 7 x + 2x – 1 Warm-Up Exercises GUIDED PRACTICE 2. (x3 – x2 + 4x – 10) for Examples 1 and 2 (x + 2) SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 x2 – 3x + 10 x + 2 )x3 – x2 + 4x – 10 quotient Multiply divisor by x3/x = x2. x3 + 2x2 –3x2 + 4x Subtract. Bring down next term. – Multiply divisor by –3x2/x = –3x. 3x2 – 6x 10x – 1 Subtract. Bring down next term. 10x + 20 Multiply divisor by 10x/x = 10. – 30 remainder Warm-Up Exercises GUIDED PRACTICE ANSWER for Examples 1 and 2 x3 – x2 +4x – 10 x+2 = (x2 – 3x +10)+ – 30 x+2 Warm-Up3Exercises EXAMPLE Use synthetic division Divide f (x)= 2x3 + x2 – 8x + 5 by x + 3 using synthetic division. SOLUTION –3 ANSWER 1 –8 5 –6 15 – 21 2 –5 7 – 16 2 2x3 + x2 – 8x + 5 16 = 2x2 – 5x + 7 – x+3 x+3 Warm-Up4Exercises EXAMPLE Factor a polynomial Factor f (x) = 3x3 – 4x2 – 28x – 16 completely given that x + 2 is a factor. SOLUTION Because x + 2 is a factor of f (x), you know that f (– 2) = 0. Use synthetic division to find the other factors. – 2 3 – 4 – 28 – 16 –6 20 16 3 – 10 –8 0 Warm-Up4Exercises EXAMPLE Factor a polynomial Use the result to write f (x) as a product of two factors and then factor completely. f (x) = 3x3 – 4x2 – 28x – 16 Write original polynomial. = (x + 2)(3x2 – 10x – 8) Write as a product of two factors. = (x + 2)(3x + 2)(x – 4) Factor trinomial. Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 Divide using synthetic division. 3. (x3 + 4x2 – x – 1) (x + 3) SOLUTION (x3 + 4x2 – x – 1) –3 1 3 ANSWER –1 4 (x + 3) –1 –3 –3 12 1 –4 11 x3 + 4 x2 – x – 1 11 = x2 + x – 4 + x+3 x+3 Warm-Up Exercises GUIDED PRACTICE 4. (4x3 + x2 – 3x + 7) for Examples 3 and 4 (x – 1) SOLUTION (4x3 + x2 – 3x + 7) 1 4 4 ANSWER (x – 1) 1 –3 7 4 5 2 2 9 5 4x3 + x2 – 3x + 1 9 2 = 4x + 5x + 2 + x–1 x–1 Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 Factor the polynomial completely given that x – 4 is a factor. 5. f (x) = x3 – 6x2 + 5x + 12 SOLUTION Because x – 4 is a factor of f (x), you know that f (4) = 0. Use synthetic division to find the other factors. 4 1 –6 5 12 4 –8 –12 1 – 2 –3 0 Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 Use the result to write f (x) as a product of two factors and then factor completely. f (x) = x3 – 6x2 + 5x + 12 Write original polynomial. = (x – 4)(x2 – 2x – 3) Write as a product of two factors. = (x – 4)(x –3)(x + 1) Factor trinomial. Warm-Up Exercises GUIDED PRACTICE 6. for Examples 3 and 4 f (x) = x3 – x2 – 22x + 40 SOLUTION Because x – 4 is a factor of f (x), you know that f (4) = 0. Use synthetic division to find the other factors. 4 1 4 – 22 40 4 1 3 12 –40 – 10 0 Warm-Up Exercises GUIDED PRACTICE for Examples 3 and 4 Use the result to write f (x) as a product of two factors and then factor completely. f (x) = x3 – x2 – 22x + 40 = (x – 4)(x2 + 3x – 10) = (x – 4)(x –2)(x +5) Write original polynomial. Write as a product of two factors. Factor trinomial. Warm-Up5Exercises EXAMPLE Standardized Test Practice SOLUTION Because f (3) = 0, x – 3 is a factor of f (x). Use synthetic division. 3 1 1 –2 – 23 60 3 3 – 60 1 – 20 0 Warm-Up5Exercises EXAMPLE Standardized Test Practice Use the result to write f (x) as a product of two factors. Then factor completely. f (x) = x3 – 2x2 – 23x + 60 = (x – 3)(x2 + x – 20) = (x – 3)(x + 5)(x – 4) The zeros are 3, – 5, and 4. ANSWER The correct answer is A. Warm-Up6Exercises EXAMPLE Use a polynomial model BUSINESS The profit P (in millions of dollars) for a shoe manufacturer can be modeled by P = – 21x3 + 46x where x is the number of shoes produced (in millions). The company now produces 1 million shoes and makes a profit of $25,000,000, but would like to cut back production. What lesser number of shoes could the company produce and still make the same profit? Warm-Up6Exercises EXAMPLE Use a polynomial model SOLUTION Substitute 25 for P in P = – 21x3 + 46x. 25 = – 21x3 + 46x 0 = 21x3 – 46x + 25 Write in standard form. You know that x = 1 is one solution of the equation. This implies that x – 1 is a factor of 21x3 – 46x + 25. Use synthetic division to find the other factors. 1 21 0 – 46 25 21 21 –25 21 21 – 25 0 Warm-Up6Exercises EXAMPLE Use a polynomial model So, (x – 1)(21x2 + 21x – 25) = 0. Use the quadratic formula to find that x 0.7 is the other positive solution. ANSWER The company could still make the same profit producing about 700,000 shoes. Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 Find the other zeros of f given that f (– 2) = 0. 7. f (x) = x3 + 2x2 – 9x – 18 SOLUTION Because f (– 2 ) = 0, x + 2 is a factor of f (x). Use synthetic division. –2 1 1 2 – 9 – 18 –2 0 18 0 –9 0 Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 Use the result to write f (x) as a product of two factors. Then factor completely. f (x) = x3 + 2x2 – 9x – 18 = (x + 2)(x2 – 92) = (x + 2)(x + 3)(x – 3) The zeros are 3, – 3, and – 2. Warm-Up Exercises GUIDED PRACTICE 8. for Examples 5 and 6 f (x) = x3 + 8x2 + 5x – 14 SOLUTION Because f (– 2 ) = 0, x + 2 is a factor of f (x). Use synthetic division. –2 1 1 8 5 – 14 –2 –12 14 6 –7 0 Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 Use the result to write f (x) as a product of two factors. Then factor completely. f (x) = x3 + 8x2 + 5x – 14 = (x + 2)(x2 + 6x – 7 ) = (x + 2)(x + 7)(x – 1) The zeros are 1, – 7, and – 2. Warm-Up Exercises GUIDED PRACTICE 9. for Examples 5 and 6 What if? In Example 6, how does the answer change if the profit for the shoe manufacturer is modeled by P = – 15x3 + 40x? SOLUTION 25 = – 15x3 + 40x 0 = 15x3 – 40x + 25 Substitute 25 for P in P = – 15x3 + 40x. Write in standard form. You know that x = 1 is one solution of the equation. This implies that x – 1 is a factor of 15x3 – 40x + 25. Use synthetic division to find the other factors. 1 15 0 15 – 40 25 15 –25 15 15 – 25 0 Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 So, (x – 1)(15x2 + 15x – 25) = 0. Use the quadratic formula to find that x 0.9 is the other positive solution. ANSWER The company could still make the same profit producing about 900,000 shoes. Warm-Up Exercises Daily Homework Quiz 1. Divide 6x4 – x3 – x2 + 11x – 18 by 2x2 + x – 3. ANSWER 3x2 –3 – 2x + 5 + 2x2 + x – 3 2. Use synthetic division to divide f(x) = x3 – 3x2 – 5x – 25 by x – 5. ANSWER x2 + 2x + 5 Warm-Up Exercises Daily Homework Quiz 3. One zero of f(x) = x3 – x2 – 17x – 15 is x = – 1. ANSWER 5 4. One of the costs to print a novel can be modeled by C = x3 – 10x2 + 28x, where x is the number of novels printed in thousands. The company now prints 5000 novels at a cost of $15,000. What other numbers of novels would cost about the same amount? ANSWER About 4300 or about 700