warm-up exercises 1. use the quadratic formula to solve 2x 2 – 3x – 1 = 0. round the nearest...

Warm-Up Exercises

1. Use the quadratic formula to solve 2x2 – 3x – 1 = 0. Round the nearest hundredth.

2. Use synthetic substitution to evaluate f (x) = x3 + x2 – 3x – 10 when x = 2.

ANSWER 1.78, –0.28

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-Up ExercisesEXAMPLE 1 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.


Multiply divisor by 3x4/x2 = 3x2

3x4 – 9x3 + 15x2

4x3 – 15x2 + 4x Subtract. Bring down next term.

Multiply divisor by 4x3/x2 = 4x4x3 – 12x2 + 20x

– 3x2 – 16x – 6 Subtract. Bring down next term.

Multiply divisor by – 3x2/x2 = – 3–3x2 + 9x – 15

– 25x + 9 remainder

3x2 + 4x – 3x2 – 3x + 5 3x4 – 5x3 + 0x2 + 4x – 6)

quotient


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

CHECK

= 3x4 – 9x3 + 15x2 + 4x3 – 12x2 + 20x – 3x2 + 9x – 15 – 25x + 9

= 3x4 – 5x3 + 4x – 6

3x4 – 5x3 + 4x – 6x2 – 3x + 5

= 3x2 + 4x – 3 + – 25x + 9x2 – 3x + 5

ANSWER

Warm-Up ExercisesEXAMPLE 2 Use polynomial long division with a linear divisor

Divide f (x) = x3 + 5x2 – 7x + 2 by x – 2.

x2 + 7x + 7x – 2 x3 + 5x2 – 7x + 2)

quotient

x3 – 2x2 Multiply divisor by x3/x = x2.

7x2 – 7x Subtract.

Multiply divisor by 7x2/x = 7x.7x2 – 14x

7x + 2 Subtract.

Multiply divisor by 7x/x = 7.16 remainder

7x – 14

ANSWER x3 + 5x2 – 7x +2x – 2

= x2 + 7x + 7 + 16x – 2

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

Divide using polynomial long division.

1. (2x4 + x3 + x – 1) (x2 + 2x – 1)

SOLUTION



Multiply divisor by 2x4/x2

= –2x2.2x4 – 4x3 – 2x2

3x3 – 2x2 + x Subtract. Bring down next term.

Multiply divisor by –3x3/x2

= –3.– 3x3 – 6x2 + 3x

8x2 – 2x – 1 Subtract. Bring down next term.

Multiply divisor by 4x2/x2 = 8.

8x2 –16x – 8

– 18x + 7 remainder

2x2 – 3x + 8x2 + 2x – 1 2x4 + x3 + 0x2 + x – 1)

quotient


2x4 + 5x3 + x – 1x2 + 2x – 1

= (2x2 – 3x + 8)+ – 18x + 7x2 + 2x – 1

ANSWER


2. (x3 – x2 + 4x – 10) (x + 2)

SOLUTION



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.

Multiply divisor by 10x/x = 10.

10x + 20

– 30 remainder

x2 – 3x + 10 x + 2 x3 – x2 + 4x – 10)

quotient


x3 – x2 +4x – 10x + 2

= (x2 – 3x +10)+ – 30 x + 2

ANSWER

Warm-Up ExercisesEXAMPLE 3Use synthetic division

Divide f (x)= 2x3 + x2 – 8x + 5 by x + 3 using synthetic division.

– 3 2 1 – 8 5

– 6 15 – 21

2 – 5 7 – 16

2x3 + x2 – 8x + 5x + 3

= 2x2 – 5x + 7 –16

x + 3ANSWER

SOLUTION

Warm-Up ExercisesEXAMPLE 4 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-Up ExercisesEXAMPLE 4 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.


Divide using synthetic division.

3. (x3 + 4x2 – x – 1) (x + 3)

SOLUTION

(x3 + 4x2 – x – 1) (x + 3)

– 3 1 4 – 1 – 1

– 3 – 3 12

3 1 – 4 11

x3 + 4 x2 – x – 1x + 3

= x2 + x – 4 +11

x + 3ANSWER


4. (4x3 + x2 – 3x + 7) (x – 1)

SOLUTION

(4x3 + x2 – 3x + 7) (x – 1)

1 4 1 – 3 7

4 5 2

4 5 2 9

4x3 + x2 – 3x + 1x – 1

= 4x2 + 5x + 2 +9

x – 1ANSWER


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



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.


6. 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 12 –40

1 3 – 10 0



f (x) = x3 – x2 – 22x + 40 Write original polynomial.

= (x – 4)(x2 + 3x – 10) Write as a product of two factors.

= (x – 4)(x –2)(x +5) Factor trinomial.

Warm-Up ExercisesEXAMPLE 5 Standardized Test Practice

SOLUTION

Because f (3) = 0, x – 3 is a factor of f (x). Use synthetic division.

3 1 – 2 – 23 60

3 3 – 60

1 1 – 20 0

Warm-Up ExercisesEXAMPLE 5

Use the result to write f (x) as a product of two factors. Then factor completely.

f (x) = x3 – 2x2 – 23x + 60

The zeros are 3, – 5, and 4.

Standardized Test Practice

The correct answer is A. ANSWER

= (x – 3)(x + 5)(x – 4)

= (x – 3)(x2 + x – 20)

Warm-Up ExercisesEXAMPLE 6 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 numberof shoes produced (in millions). The companynow 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?


SOLUTION

25 = – 21x3 + 46x Substitute 25 for P in P = – 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


So, (x – 1)(21x2 + 21x – 25) = 0. Use the quadratic formula to find that x 0.7 is the other positive solution.

The company could still make the same profit producing about 700,000 shoes.

ANSWER


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 2 – 9 – 18

– 2 0 18

1 0 – 9 0



f (x) = x3 + 2x2 – 9x – 18

The zeros are 3, – 3, and – 2.

= (x + 2)(x + 3)(x – 3)

= (x + 2)(x2 – 92)


8. f (x) = x3 + 8x2 + 5x – 14

SOLUTION

Because f (– 2 ) = 0, x + 2 is a factor of f (x). Use synthetic division.

– 2 1 8 5 – 14

– 2 –12 14

1 6 – 7 0



f (x) = x3 + 8x2 + 5x – 14

The zeros are 1, – 7, and – 2.

= (x + 2)(x + 7)(x – 1)

= (x + 2)(x2 + 6x – 7 )


9. 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 Substitute 25 for P in P = – 15x3 + 40x.0 = 15x3 – 40x + 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 15x3 – 40x + 25. Use synthetic division to find the other factors.

1 15 0 – 40 25

15 15 –25

15 15 – 25 0


So, (x – 1)(15x2 + 15x – 25) = 0. Use the quadratic formula to find that x 0.9 is the other positive solution.

The company could still make the same profit producing about 900,000 shoes.

ANSWER

Warm-Up ExercisesDaily Homework Quiz

2. Use synthetic division to divide f(x) = x3 – 3x2 – 5x – 25 by x – 5.

1. Divide 6x4 – x3 – x2 + 11x – 18 by 2x2 + x – 3.

ANSWER

3x2 – 2x + 5 + 2x2 + x – 3– 3

ANSWER

x2 + 2x + 5

Warm-Up ExercisesDaily Homework Quiz

ANSWER

5

ANSWER

About 4300 or about 700

3. One zero of f(x) = x3 – x2 – 17x – 15 is x = – 1.

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?

warm-up exercises 1. use the quadratic formula to solve 2x 2 – 3x – 1 = 0. round the nearest...

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