Understanding Advanced Factoring. Factor Theorem For polynomial P(x), (x - a) is a factor of P(x) if and only if P(a) = 0.

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Warm Up

61

L E S S O N

1. Vocabulary A polynomial of degree one with two terms is a . 2. True or False: The GCF of -2x4_y5_-_xy2₊_{2 is 2xy}2_.

3. Divide 4x2_-_2x₊_{5 by x}_-_{1 using synthetic substitution.}

Recall that synthetic division is used to divide a polynomial P(x) by a linear binomial of the form x - a and that the last number in the bottom row is the remainder and also the value of P(a). If the remainder is 0, the linear binomial is a factor of the polynomial. This is known as the Factor Theorem.

Factor Theorem

For polynomial P(x), (x - a) is a factor of P(x) if and only if P(a)= 0. For example, x + 2 is a factor of x2₊_10x₊_{16 and}

P(-2)=(-2)2₊₁₀_(-₂₎₊₁₆ = 4 - 20 + 16

= 0.

Example 1 Determining Whether a Linear Binomial is a Factor

Determine whether the linear binomial is a factor of the polynomial.

a. P(x)= 2x3 - x2

- 43x + 60, x - 4 SOLUTION Use synthetic division with a = 4.

4 2 -1 -43 60

8 28 -60

2 7 -15 0

Because P(4)= 0, x - 4 is a factor of P(x)= 2x3_-_x2_-_43x₊_60.

b. P(x)= x3₊_x2_-_24x₊_{36, x}₊₃

SOLUTION Use synthetic division with a =-3.

-3 1 1 -24 36

-3 6 54

1 -2 -_{18 90} Because P(-3)≠ 0, x + 3 is not a factor of P(x)= x3

+ x2

- 24x + 36. When the remainder is zero, the quotient of P(x) and (x - a) is another factor of the polynomial. This factor may be able to be further factored by other factoring methods. For instance, in Example 1a, the quotient

2x2₊_7x_-_{15 factors into}₍_2x_-₃₎₍_x₊₅₎_{. Therefore, P}₍_x₎_{factors into}

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New Concepts

Understanding Advanced Factoring

Online Connection

Hint

x+ 2 =x-(-2), so the value of a is -2.

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Lesson 61 437

If the remaining quotient is a cubic, it may be factorable by one of the following factoring methods.

Sum and Difference of Cubes Sum of Two Cubes: a3

+ b3

=(a + b)(a2

- ab + b2 )

Difference of Two Cubes: a3 - b3

=(a - b)(a2

+ ab + b2 )

Example 2 Factoring the Sum and Difference of Two Cubes

Factor each expression.

a. d 3₊₁₂₅

SOLUTION The expression is a sum of two cubes, where a = d and b = 5.

d 3₊₁₂₅

(d + 5)(d 2_-_5d₊₂₅₎

b. 27x6+ y9

SOLUTION The expression is a sum of two cubes, where a = 3x2_{and b}₌_y3_.

27x6₊_y9₌₍_3x2₎3₊₍_y3₎3 (3x2₊_y3₎₍_9x4_-_3x2_y3₊_y6₎

c. 3m5_-_24m2

SOLUTION First factor out the GCF, 3m2_{. Then factor the difference of the}

two cubes, where a = m and b = 2. 3m5_-_24m2

3m2₍_m3_-₈₎

3m2₍_m_-₂₎₍_m2₊_2m₊₄₎

Another method of factoring is factoring by grouping. To factor by grouping, group the terms evenly so that each group has a common factor. Factor out the GCF from each group. Then factor out the common polynomial factor.

Example 3 Factoring by Grouping

Factor each expression.

a. x3₊_5x2₊_3x₊₁₅

SOLUTION The first two terms have a common factor as do the last two terms.

(x3₊_5x2₎₊₍_3x₊₁₅₎

x2₍_x₊₅₎₊₃₍_x₊₅₎_Factor_x2_{from the first group and} 3 from the second.

(x + 5)(x2₊₃₎ _{Factor out the GCF of}₍_x₊₅₎_.

Neither factor can be factored further.

Math Reasoning

Justify Tell why 3x- 1 is not a difference of cubes.

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b. x4+ 4x3

- x - 4

SOLUTION Factor x3_{from the first two terms and}

-1 from the last two terms.

x4 + 4x3 - x - 4 x3 (x + 4)- 1(x + 4) (x + 4)(x3

- 1) Factor out the GCF of (x+4). (x + 4)(x - 1)(x2

+ x + 1) Factor the difference of cubes.

c. x6 + 3x5

+ 2x4 + x2

+ 3x + 2

SOLUTION Use two groups of three terms each.

(x6 + 3x5 + 2x4 )+(x2 + 3x + 2) x4 (x2 + 3x + 2)+ 1(x2 + 3x + 2) (x2 + 3x + 2)(x4

+ 1) Factor out the GCF of (x2

+3x+2). (x + 1)(x + 2)(x4

+ 1) Factor the trinomial.

Example 4 Application: Three-Dimensional Figures

The volume of a rectangular solid with a length of x - 5 feet has a volume of x3_-_23x2₊_170x_-_{400 cubic feet. Factor the expression for the volume}

completely.

SOLUTION Since volume is the product of the length, width, and height, the expression for the length, x - 5, is a factor of the polynomial. Use synthetic division to find the quotient of the polynomial and x - 5.

5 1 -23 170 -400 5 -90 400

1 -18 80 0

The expression for the volume factors into (x - 5)(x2_-_18x₊₈₀₎_{. But the}

trinomial can still be factored: (x - 10)(x - 8).

The expression for the volume factors into (x - 5)(x - 10)(x - 8).

Lesson Practice

Determine whether the linear binomial is a factor of the polynomial. a. P(x)= 3x3₊_19x2₊_32x₊_{16, x}₊₃

b. P(x)= 4x3_-_33x2₊_56x_-_{12, x}_-₆

Factor each expression.

c. 64 - h3 _{d. s}6_t12 _-_125r3

e.

_

8 27 x

7₊_x4 _{f. xy}_-_9y₊_5x_-₄₅

g. 4m2_n₊_12m2_-_n_-₃ _{h. x}5₊_12x4₊_36x3₊_5x2₊_60x₊₁₈₀

i. The volume of a rectangular solid with a length of x - 15 feet has a volume of x3₊_5x2_-_600x₊_{4500 cubic feet. Factor the expression for}

the volume completely. (Ex 1)

(Ex 1) (Ex 1) (Ex 1)

(Ex 2)

(Ex 2) (Ex 2)(Ex 2)

(Ex 2)

(Ex 2) (Ex 3)(Ex 3)

(Ex 3)

(Ex 3) (Ex 3)(Ex 3)

(Ex 4) (Ex 4)

Math Reasoning

Analyze Can a polynomial with five terms be factored by using the grouping method? Explain.

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Practice

Distributed and Integrated

Lesson 61 439

Graph.

1. x =-5 2. 2x - y = 4 3. 7y + 7x = 21

4. Find the intercepts of the equation 5x - 4y + 7z = 21. Simplify. 5. - 16 - _ 12 _{6. 27}- 1 _ 3 _{7. 9} 3 _ ₂ Find an inverse for the equations.

8. y =

_

2

3 x - 3 9. y = 5x + 15

10. Let K(u)= 17 and L(u)=-13. Find K

(

L(u)) and L

(

K(u)) .

*11. Minimize P = 4x + 3y for the constraints

x ≤ 4 y ≤ 6 x + y ≥ 7

*12. Mary deposited $980 at 7.0% compounded continuously. How much money will she have after 9 years?

13. Meteorology The number of tornadoes in the United States in 2006 is shown in the table.

Season Jan-Feb Mar-Apr May-Jun Jul-Aug Sep-Oct Nov-Dec

Number of Tornadoes 59 395 258 151 160 82

What is the experimental probability that a tornado occurs in the first six months of the year?

*14. Write Tell how factoring a sum of cubes is similar to factoring a difference of cubes. 15. Probability A student graphed a set of data points and found that the correlation

coefficient was -0.73. What is the probability that the slope of the line of best fit is greater than _ 1₂ ? Explain.

*16. Geometry The probability that a randomly selected point lies within a region can be found using the ratio of the area the point would lie in to the total area. Find the probability that a random point within the region lies within one of the smaller circles.

(13) (13) (13)(13) (13)(13) (Inv 3) (Inv 3) (59) (59) (59)(59) (59)(59) (50) (50) (50)(50) (53) (53) (54) (54) (57) (57) (55) (55) (61) (61) (45) (45) 6 in. 10 in. 2 in. 6 in. 10 in. 2 in. 6 in. 10 in. 2 in. (60) (60)

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17. Buildings One building is 310 feet tall and a nearby building is 374 feet tall. From the top of the taller building, you can spot the top of the shorter building at an angle of depression of 30°. How far apart are the two buildings? Round to the nearest foot.

18. Walking Shadows While walking the dog one night, you notice how the length of your shadow changes as you approach each lamppost. The length of the shadow seems to be _ 1₅ of your distance to the lamppost. Your dog is dragging you along at about 3 feet per second, aiming for a lamppost. Show how to write a composite function that gives the length of your shadow in terms of time as you approach the lamppost.

*19. Error Analysis Is the work below correct? If not, find and explain the error, and give the correct solution.

x _ 38 ₌₃

(

_x 3 _ ₈

)

3 _ ₈ = (3) 3 _ ₈ x1₌₃ _ 3₈ x = √ 8 27 20. Analyze Suppose you spin the spinner shown 8 times. What is the probability

that you will spin black 5 times?

21. Verify Use the form x2

+ 2cx + c2

for a quadratic perfect square trinomial to verify the method of completing the square.

*22. Graphing Calculator Describe the feasible region for the inequalities:

x ≥ 0, y ≥ 0, y ≤-0.75x + 8

23. Multiple Choice Which value of r, the correlation coefficient, describes the set of points that are closest to forming a line?

A r =-0.75 B r = 0 C r = 0.6 D r = 0.65

*24. Commuting to Work According to data collected in the 2005 American Community Survey, approximately 10.67% of workers carpool to work, 4.66% use public transportation, and 2.47% walk. Find the probability that a randomly selected worker carpools or uses public transportation.

25. Write A few different equations of the form y = b x_{are graphed to the} right. Describe in your own words how the shape of the graph depends on the value of b. (46) (46) (53) (53) (59) (59) (49) (49) (58) (58) (54) (54) (45) (45) (60) (60) x y O 4 6 2 2 4 -2 -2 -4 y=.5x y=2.5 x y=1.25x x y O 4 6 2 2 4 -2 -2 -4 y=.5x y=2.5 x y=1.25x (47) (47)

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Lesson 61 441 * 26. Multi-Step Use 5 3 _ ₅ · 5 3_ 5

_

5 _ 45 .

a. Which property of rational exponents should be applied first to simplify the expression? Explain.

b. Apply the property from part a.

c. Which property of rational exponents should be applied next? d. Apply the property from part c.

e. If possible, finish simplifying the expression from part d. 27. Multiple Choice In the diagram to the right, find the length of the

hypotenuse of triangle LMN.

A 7.5 B √ 2

C 15 D 15 √ 3

*28. Multi-Step The area of a twin size mattress is x2_-₂₇₅_-_11x₊_{25x square inches.}

a. Factor the polynomial by grouping. b. Find the area given that x = 50.

29. Amusement Parks The Giant Swing at Silver Dollar City in Branson, Missouri, rotates through an angle measuring 230°. Find the reference angle for this angle.

*30. Verify Show that there are two different ways to group the terms in 3d + 6f + d2₊_{2df and that both lead to the same factorization.}

(59) (59) L M N 15 _ √2cm L M N 15 _ √2cm (52) (52) (61) (61) (56) (56) (61) (61)

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Warm Up

62

L E S S O N

1. Vocabulary Any point on the number line is a number. 2. Simplify (5 + 3x)-(6 - 2x).

3. Solve 5x2_-₁₂₅₌_0.

4. Simplify √ 150 .

The equation x =± √ -16 has no real solutions because there are no real numbers that, when squared, equal -16. But the equation does have solutions when a new set of numbers, outside of the real numbers, is introduced. These numbers are the imaginary numbers.

Imaginary Numbers

An imaginary number is written in the form bi where b is a real number and i is the imaginary unit.

i is a solution of x2₌_-_1.

Because i2₌_-_{1, i}₌_√_-_{1 .}

The square root of a negative number is an imaginary number. √ -16 = √ 16 ·-1 = √ 16 √ -1 = 4i

Example 1 Simplifying Square Roots of Negative Numbers

a. Simplify -2 √ -49 .

SOLUTION

-2 √ - 49

-2 √ 49 ·-1 Write -49 as the product of 49 and -1.

-2 √ 49 √ -1 The square root of a product is the product of the square roots.

-2 · 7 · i √ 49 = 7 and √ -1 = i. -14i Multiply the real numbers.

b. Simplify √ -117 . SOLUTION

√ -117

√ 9 · 13 ·-1 Write -117 as a product with a perfect square.

√ 9 √ 13 √ -1 The square root of a product is the product of the square roots. 3 √ 13 (i) √ 9 = 3 and √ -1 = i. (1) (1) (2) (2) (58) (58) (40) (40)

New Concepts

Using Complex Numbers

Math Reasoning

Generalize For any positive real n,

√-n = .

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Lesson 62 443

Example 2 Solving a Quadratic Equation with Imaginary

Numbers

Solve x2₊₂₅₌_{0. Write the solutions in terms of i. Check the answers.}

SOLUTION x2₊₂₅₌₀

x2₌_-₂₅ _{Subtract 25 from each side.}

x= ± √-25 Take the square root of each side.

x= ±5i Simplify the square root.

The solutions are the conjugate pairs 5i and -5i. Check x2₊₂₅₌₀ (5i)2₊₂₅₀ 25i2₊₂₅₀ 25(-1)+ 25 0 0 = 0

✓

x2₊₂₅₌₀ (-5i)2₊₂₅₀ 25i2₊₂₅₀ 25(-1)+ 25 0 0 = 0

✓

An imaginary number bi is part of a complex number.

Complex Numbers

A complex number is a number that can be written in the form a + bi,

where a and b are real numbers.

In a complex number, a is called the real part and bi is called the

imaginary part. If b = 0, then the imaginary part is 0 and the number is a real number. If a = 0 and b ≠ 0, then the real part is 0 and the number is an imaginary number.

Complex Real Imaginary

2 + 3i 2 + 0i = 2 0 + 3i = 3i

To add or subtract complex numbers, add or subtract the real parts and then add or subtract the imaginary parts.

Add or Subtract Complex Numbers

(a + bi) + (c + di) = (a + c) + (b + d )i (a + bi) - (c + di) = (a - c) + (b - d )i

Example 3 Simplifying Expressions with Complex Numbers

Add or subtract. Write the answer in the form a + bi.

a. (7 + 4i) + (-2 - 5i)

SOLUTION (7 + (-2)) + (4 + (-5))i = 5 + (-1)i = 5 - i

b. (5 - 3i)-(5 - 11i)

SOLUTION(5 - 5)+(-3 -(-11))i = 0 + 8i = 8i

Math Reasoning

Analyze Are real numbers a subset of complex numbers or are complex numbers a subset of real numbers?

Math Language

Complex solutions of quadratic equations always occur in conjugate pairs of the form a+ bi and a-bi.

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Practice

Example 4 Solving Quadratic Equations

Solve each equation. Write the solutions in the form a + bi.

a. (x - 3)2_{= -}₄

SOLUTION (x - 3)2_{= -}₄

x - 3 = ± √ -4 Take the square root of each side.

x - 3 = ±2i Simplify the square root.

x - 3 = 2i or x - 3 = -2i Solve both equations.

x = 3 + 2i or x = 3 - 2i The solutions are 3 + 2i and 3 - 2i.

b. x2 + 8x + 18 = 0

SOLUTION x2₊_8x_{= -}₁₈ _{Subtract 18 from each side.}

x2₊_8x₊₁₆_{= -}₁₈₊₁₆ _{Complete the square.}

(x + 4)2 _{= -}₂ _{Write the left side as a perfect square.}

x + 4 = ± √ -2 Take the square root of each side.

x + 4 =±i √ 2 Simplify the square root.

x = -4 ± i √ 2 Subtract 4 from each side.

The solutions are -4 + i √ 2 and -4 - i √ 2 .

Lesson Practice

a. Simplify _ 1₂ √ -100 . b. Simplify 3 √ -450 .

c. Solve 0 = 196 + x2_{. Write the solutions in terms of i. Check the answer.}

d. Solve -6x2₌_{216. Write the solutions in terms of i.}

Add or subtract. Write the answer in the form a + bi. e. (-2 + i) + (14 + 4i)

f. (10 - 6i) - (-4 + 3i)

Solve each equation. Write the solutions in the form a + bi. g. (x + 9)2_{= -}₉ h. x2_-_2x₊₂₌₀ (Ex 1) (Ex 1) (Ex 1) (Ex 1) (Ex 2) (Ex 2) (Ex 2) (Ex 2) (Ex 3) (Ex 3) (Ex 4) (Ex 4) Factor. *1. 125x4 + 375x3 - x - 3 *2. 64x12 - 125y9 (61) (61) (61)(61)

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_

1 81 _ -34 9. (-27 ) _ -23 _10.

_

1 -3 -2

11. School It is estimated that 15% of students walk to school. What is the probability that exactly 2 out of 5 randomly selected students will walk to school?

12. Find an equation for the inverse of y = _ 1 5 x

3₊_6.

13. What are the odds in favor of rolling a 3 on a number cube? *14. Multiple Choice Which is a real number?

A 15i B 0 + 15i C 5 + 0i D 15 + 15i

*15. Verify Show that the expression _ 1

1 + _ _x_{+ 1}1 is equivalent to x + 1

_ _x_{+ 2}.

*16. Electrical Engineering A circuit has a current of (4 - 2i) amps, and a second circuit has a current of (6 - 6i) amps. Find the sum of the currents.

17. Analyze Explain why the odds in favor of an event with a probability of 50% are 1 to 1. 18. Projectile Motion The Leaning Tower of Pisa measures 180.4 feet. If an object is

thrown into the air, with an initial velocity of 80 feet per second, from the top of the lower side, the height, h, of the object in feet after t seconds can be described by the equation h = -16t2₊_80t₊_{180.4. Find the time at which the object will}

reach a height of 200 feet. Round your answer to the tenths place.

*19. A number cube is rolled once. Are the events a number greater than 3 or a multiple of 2 mutually exclusive or inclusive? Explain why.

*20. Transportation The approximate volume of a moving truck, in cubic feet, can be given by x3_-_6x2₊_8x₊_5x2_-_30x₊_{40. Factor this expression by grouping.}

Then find the volume when x = 10.

21. Multiple Choice Which of the following angles is not coterminal with 225º?

A 585º B -135º C 935º D -495º

22. Multi-Step The population of an insect colony is growing at a rate of 1.5% per week. a. Write an expression for the function N(P) that gives the population one week

after the population is P.

b. What composite function gives the population three weeks after the population is P? (37) (37) (37)(37) (37)(37) (23) (23) (23)(23) (59) (59) (59)(59) (59)(59) (49) (49) (50) (50) (55) (55) (62) (62) (37) (37) (62) (62) (55) (55) (58) (58) (60) (60) (61) (61) (56) (56) (57) (57)

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*23. Graphing Calculator Factor 2x5_-_x4_-_2x₊_{1. Then enter the original expression for}

Y1 and the factored expression for Y2 on a graphing calculator. Access the Table function and study the table of values for the expressions. What is true? What does it mean?

24. Write Explain why radical expressions with even indices and positive radicands have two real roots. Use √ 664 to demonstrate your explanation.

*25. Geometry An expression for the volume of the rectangular prism is V = 2x3₊_7x2_-_14x_-_{40. Find an expression for}

the missing dimension.

26. Analyze Suppose the graph of y = b x₊_{k passes through the} points (3, -3) and (5, 0). Determine whether the equation models exponential growth or decay.

27. Error Analysis A student made an error while studying the relation between the two triangles shown. Explain the error.

Find the missing angles.

m∠Y = 90° - 30° = 60° m∠R = 90° - 60° = 30° m∠X = 90° m∠Y = 60° m∠Z = 30°

m∠P = 90° m∠Q = 60° m∠R = 30°

The corresponding angles are congruent. So, the triangles are congruent.

28. Probability A sales clerk has a weekly quota of 15 sales. The probability, p, that the sales clerk will meet this quota after working d days, in a 5-day work week, can be approximated by p = -0.04d2₊_{0.4d. How many days a week should the sales}

clerk work for the probability of making 15 sales to be 0.8? *29. Verify Show that 11i and -11i are solutions of x2_{= -}_121.

30. Cell Phones Mobile-U has two cell phone plans, the Metropolitan Plan and the Continental Plan. The cost for each plan is shown in the table. A company is comparing the two plans. The company needs at least 12 cell phones, one for each sales representative. Each representative will use at least 500 minutes per month. What options does this company have?

Continental Plan Metropolitan Plan

Start-up Fee $50 $100

Free minutes 2000 1200

Per minute rate above free minutes

10 cents 7 cents

Monthly

charge $20 × # of cell phones $25 × # of cell phones

(61) (61) (59) (59) (51) (51) x+ 2 x+ 4 ? x+ 2 x+ 4 ? (57) (57) (52) (52) X Y Z P Q R 30° 60° X Y Z P Q R 30° 60° (58) (58) (62) (62) (54) (54)

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Lesson 63 447

63

L E S S O N

1. Vocabulary Half the diameter of a circle is the of the circle. 2. If both legs of a right triangle have length 1, then the length of the

hypotenuse is .

3. If the shortest leg of a 30°–60°–90° triangle has length 1, then the lengths of the other two sides of the triangle are and .

A unit circle is a circle with a radius of 1 unit.

Exploration

Exploration Exploring the Unit Circle

Use trigonometric ratios to determine the coordinates of points on a unit circle that is centered at the origin.

1. The figure shows a unit circle and a 60° angle in standard position. What is the length of the hypotenuse of the 30°-60°-90° triangle? How do you know?

2. Use what you know about special right triangles to find the lengths of the legs of the 30°-60°-90°

triangle.

3. What are the coordinates of point P?

4. What are the exact values of cos 60° and sin 60°?

5. How are the values of cos 60° and sin 60° related to the coordinates of point P?

6. Describe how you can use a similar method to find the coordinates of point Q.

7. Explain how the values of cos 45° and sin 45°

are related to the coordinates of point Q.

Angles can be measured in degrees or in radians.

The measure of an angle in radians is based on arc length, the distance between two points on a circle. In a circle with radius r, the measure of a central angle θ is one radian when it intercepts an arc that has a length equal to the radius. (SB) (SB) (41) (41) (52) (52)

New Concepts

x y x y P(x,y) 1 60° x y x y P(x,y) 1 60° x y x y Q(x,y) 1 45° x y x y Q(x,y) 1 45° r r θ= 1 radian r r θ= 1 radian

Understanding the Unit Circle and

Radian Measures

Warm Up

www.SaxonMathResources.com

Reading Math

Remember the symbol θ

is read “theta.”

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There are 360°or 2π radians in a circle. Thus the conversion

factor _ _360°2π , which is equivalent to 1, can be used to convert measures between degrees and radians.

All angles are measured clockwise or counterclockwise from the positive

x-axis. Measured clockwise, they are

negative angles. In the figure, the

-210° angle is equivalent to an angle measure of 150°. Converting Angle Measures

Degrees to Radians Radians to Degrees

Multiply the number of degrees by

(

_ π radians_180°

)

.

Multiply the number of radians by

(

_ _π_radians180°

)

.

Example 1 Converting Between Degrees and Radians

Convert each measure from degrees to radians or from radians to degrees. a. -30° SOLUTION -30°

(

3π 4 radians

)

(

180°

_

_π radians

)

= 135° Multiply by

(

180°

_

πradians

)

.

For every point P(x, y) on the unit circle, the value of

(cos θ, sin θ).

_{_}√2 _{_}√2 2 , 2

)

(

_1 2 , √3 _ 2

)

The Unit Circle

II III IV I

(

_{_}√2 2 , √2 _ 2

)

)

(

_1 2 , √3 _ 2

)

The Unit Circle

II III IV I

(

_{_}√2 2 , √2 _ 2

)

π x y θ=360°=2πradians θ= -210°_{= -}7π 6 Math Language Angle measures in radians often appear without the label radians. For example, - _ π₆ radian is often written simply as - _ π₆ .

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Lesson 63 449

Example 2 Using the Unit Circle to Evaluate Trigonometric

Functions

Use the unit circle to find each trigonometric function value. Find exact values.

a. sin 120°

SOLUTION

You can use reference angles and the portion of the unit circle in Quadrant I to determine trigonometric function values.

Evaluate Trigonometric Functions using Reference Angles To find the sine, cosine, or tangent of θ:

Step 1: Determine the measure of the reference angle of θ.

Step 2: Use the portion of the unit circle in Quadrant I to find the sine, cosine, or tangent of the reference angle.

Step 3: Use the quadrant of the terminal side of θ in standard position to determine the sign of the sine, cosine, or tangent.

Hint

The exact value of sin 120° is irrational. An approximate value is 0.866. sin 120° = _ √ 3 2 ≈ 0.866 exact approximate

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The diagram shows how the signs of the trigonometric functions are determined by the quadrant that contains the terminal side of θ in standard position.

Example 3 Using Reference Angles to Evaluate Trigonometric

Functions

Find the sine, cosine, and tangent of 135°. Find exact values. SOLUTION

Step 1: Find the measure of the reference angle. The measure of the reference angle is 45°.

Step 2: Find the sine, cosine, and tangent of the reference angle.

sin 45° =

_

√ 2

2 Use sin θ = y. cos 45° =

_

√ 2

2 Use cosθ = x. tan 45° = 1 Use tanθ =_yx .

Step 3: Determine the sign. sin 135° =

_

√ 2

2 In Quadrant II, sin θ is positive. cos 135° = -

_

√ 2

2 In Quadrant II, cosθis negative. tan 135° = -1 In Quadrant II, tanθ is negative.

The arc length of a circle intercepted by the central angle is related to the central angle.

Arc Length Formula For a circle of radius r, the arc length s intercepted by a central angle θ(measured in radians) is given by the following formula.

s= rθ

The arc length s of a circle is a portion of the circumference of a circle. This portion is the ratio between the measure of the central angle and the measure of the entire circle multiplied by the circumference of the circle. Thus,

s = _ _2πθ · 2πr or s =θr. QII QIII QIV QI sinθ + sinθ + cosθ cosθ + tanθ tanθ + sinθ sinθ cosθ cosθ + tanθ + tanθ QII QIII QIV QI sinθ + sinθ + cosθ cosθ + tanθ tanθ + sinθ sinθ cosθ cosθ + tanθ + tanθ x y 45° 135° x y 45° 135° x y 45°

(

_{_}√₂ 2 , √₂ _ 2

)

x y 45°

(

_{_}√₂ 2 , √₂ _ 2

)

r s θ r s θ Hint

Recall that a reference angle is an acute angle. So for θ < 360°, the measure of θ is: QII - 180°- θ QIII - θ - 180° QIV - 360°- θ Math Language An arc is an unbroken part of a circle consisting of two points on the circle, called endpoints, and all the points on the circle between them. This arc is named RS .

R S

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Lesson 63 451

Example 4 Finding Arc Lengths

a. Find the length of arc s1. Approximate to the nearest tenth.

SOLUTION

s = rθ Write the formula.

s1 = 5 ·

4 radians

Step 2: Use the formula for arc length.

s = rθ Write the formula.

s2 = 2.5

(

5π

_

4

)

Substitue s2 for s, 2.5 for r, and 5π

_

4 for θ.

= 3.125π Simplify.

≈ 9.8 ft Use a calculator to approximate.

Example 5 Application: Planet Rotation

Earth makes one complete

rotation in 24 hours, and its radius is approximately

3955 miles. If an object is fixed on Earth’s equator, how far does it travel in 1 hour due to Earth’s rotation?

SOLUTION

The central angle that corresponds to a complete rotation is 2π radians. In 1 hour, Earth makes _ 1

24 of a complete rotation. So, θ = 1 _ 24 · 2π = π _ 12 . s = rθ = 3955 ·

_

π 12 ≈ 1035

The object travels about 1035 miles in 1 hour.

Check Find the approximate circumference of Earth: 2πr ≈ 2π· 3955 ≈

24,850 miles.

_ ₂₄1 of 24,850 is slightly greater than 1000, so 1035 miles is reasonable. 5 cm s1 2π _ 3 5 cm s1 2π _ 3 2.5 ft 225° s2 2.5 ft 225° s2 1 _ 24of a rotation in 1 hour r= 3955 mi 1 _ 24of a rotation in 1 hour r= 3955 mi

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Practice

Lesson Practice

a. Convert 150° to radians. b. Convert - _ 4₃π radians to degrees.

c. Use the unit circle to find the exact value of cos 315°. d. Use the unit circle to find the exact value of tan _ 3₄π .

e. Use a reference angle to find the sine, cosine, and tangent of 210°. Find exact values.

f. Find the length of arc s1. Approximate to the nearest

tenth.

g. Find the length of arc s2. Approximate to the nearest

tenth.

Expand. 8. (x + 5)3 _9.₍_x₊₄₎3 (48) (48) (48)(48) (48)(48) (Inv 5) (Inv 5) (Inv 5) (Inv 5) (40) (40) (40)(40) (19) (19) (19)(19)

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Lesson 63 453

Solve.

10. 36x2_-₃₆₌₀ _{11. 24x}_{= -}_11x2_-_x3

12. Error Analysis Julian is finding the inverse of the function y = 2x3_{. Which step is}

incorrect?

Step One: y = 2x3 _{Step Two: x}₌_2y3 _{Step Three:}

_

x

2 = y

3

Step Four: ±

√

_

x 2 = y

*18. Graphing Calculator Describe the feasible region for the set of inequalities x ≥ 0,

y ≥ 0, y ≤ -1.5x + 9.5.

*19. Formulate A fair coin is flipped. Write the equation that can be used to solve for the probability that three flips result in heads, then four flips, and seven flips. Write a formula using exponents that can be used to find the probability that the same independent event will occur n times.

20. Multiple Choice Which of the following expressions is equivalent to 1 _ _x + _ 1_y _ 1 - _ 1x ? A

_

1 - x x + y B x2_-_xy_-_x_-_y

__

x2_y C x + y

_

_xy_-_y D

_

2(1 - x) x + y

*21. Write How is adding and subtracting complex numbers similar to adding and subtracting polynomials?

*22. Model Sketch a graph of five points whose correlation coefficient is about 0.1. (35) (35) (35)(35) (50) (50) (62) (62) (44) (44) B A C 45° 45° 90° 9 m x m x m B A C 45° 45° 90° 9 m x m x m (63) (63) (57) (57) (46) (46) (54) (54) (60) (60) (48) (48) (62) (62) (45) (45)

(19)

23. Multi-Step The formula θ = _ 180°(n_n - 2) gives the measure of each interior angle of an n-sided regular polygon.

a. Use the formula to find the measure of an interior angle of a regular decagon.

b. Find the reference angle for this angle.

24. Estimate Use what you know about the unit circle and special angles to estimate cos(0.3π). Do not use a calculator. Explain your method.

25. Jupiter Gravity on the planet Jupiter is almost three times stronger than on Earth. If an object is thrown into the air, with an initial velocity of 100 feet per second, from the surface of Jupiter, the height, h, of the object in feet after t seconds can be approximated by the equation h = -42t2₊_{100t. Find the time at which the object}

will first reach 50 feet. Round your answer to the tenths place.

26. Analyze Let f(x) = ax + b and g(x) = cx + d. Under which conditions will the

composite functions f(g(x)) and g(f(x)) be equal?

27. Construction The volume of a typical American-made brick in cubic millimeters can be represented by the polynomial x3_-_20x2_-_4500x₊_{126,000 where the}

length is given by x + 70. Fully factor the expression for the volume of a brick. 28. Coordinate Geometry Sketch a graph of the equation y = 3

√ x , where y is the side

length of a cube and x is the volume of the cube. Use the graph to estimate the volume of a cube with side lengths of 2.4 in. Use the graph to estimate the side length of a cube with a volume of 6.9 in3_.

*29. Error Analysis A student incorrectly found s, the length of the indicated arc, as shown below.

s = rθ

= 10 · 60 = 600 m

What is the error? Find the correct arc length.

*30. Pass codes A computer generates temporary pass codes for users. Each pass code is exactly 3 letters long with no repeating letters. What is the probability of getting a pass code with 3 vowels (A, E, I, O, or U)?

(56) (56) (63) (63) (58) (58) (53) (53) (61) (61) (59) (59) (63) (63) 10 m 60° s 10 m 60° s (55) (55)

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Lab 10 455

L A B

10

Evaluating Logarithmic Expressions

1. Calculate the common logarithmic expression 4 log (5).

a. To enter the value 4 log (5), press .

b. To calculate this value, press .

2. Calculate the natural logarithmic expression 4 ln (5).

a. To enter the value 4 ln (5), press .

b. To calculate this value, press . 3. Use the Change of Base Law to calculate the

logarithmic expression log6 (15).

a. To enter the value log6 (15), press

.

b. Press to calculate this value.

Graphing Logarithmic Functions

1. Graph the common logarithmic function log (3x + 2).

a. Press to access the screen to enter

the logarithmic function.

b. To enter the function log (3x + 2), press .

c. Press , and then press to select

the ZStandard window.

d. Then press to view the graph.

Using the Log Keys

Graphing Calculator Lab (Use with Lesson # 64, 72, 81, 87, 93, 102, and 110)

Online Connection www.SaxonMathResources.com Graphing Calculator Tip When graphing logarithmic expressions, use the standard zoom window. Adjust the window to view specific parts of the graph by zooming in or out.

Hint

Use the Change of Base Law when calculating a log that does not have a base of 10 or e.

SM_A2_NLB_SBK_Lab10.indd Page 455 5/14/08 1:19:47 PM user

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2. Graph the logarithmic function log6(13x)

whose base is not natural or common.

a. Press to access the screen to enter

the logarithmic function.

b. To enter the function log6(13x), press

.

c. Press , and then press to select

the ZStandard window.

d. Then press to view the graph.

Lab Practice

1. Calculate 6 log (8). 2. Calculate 3 ln (1). 3. Calculate log7(9). 4. Graph y = log (15x + 8). 5. Graph y = log5(x + 6). SM_A2_NLB_SBK_Lab10.indd Page 456 4/14/08 2:07:47 PM elhi1

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Lesson 64 457

Using Logarithms

Warm Up

64

L E S S O N

1. Vocabulary In the expression 45_{, the base is 4 and the} _{is 5.}

2. What is the value of 23_?

3. What value of x makes the equation 10x₌_{100 true?}

A logarithm is the exponent that is applied to a specified base to obtain a given value. Every exponential equation has a logarithmic form and vice versa.

Exponential Equation Logarithmic Equation

b x₌_{a log} b a = x

b > 0, b ≠ 1

The logarithmic equation logb a = x is read “the log base b of a equals x.” Notice that x is both the exponent and the logarithm in the equations above.

Example 1 Converting from Exponential to Logarithmic Form

Write each exponential equation in logarithmic form.

a. 25 = 32 SOLUTION 25

= 32 log2 32 = 5 The base is the same in both the exponential

equation and the logarithmic equation.

The logarithmic form is log232 = 5.

b. 51 = 5 SOLUTION 51

= 5 log55 = 1 The exponent is the logarithm.

c. 80 = 1 SOLUTION 80

= 1 log81 = 0 Any nonzero base to the zero power is 1.

(3) (3) (3) (3) (47) (47)

New Concepts

Online Connection www.SaxonMathResources.com Reading Math

Read the symbol as “if and only if.” When used between two equations, it means the equations are equivalent.

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d. 5-2 = 0.04 SOLUTION 5-2₌_0.04_log

5 0.04 = -2 An exponent (or log) can be negative.

The logarithmic form is log5 0.04 = -2.

Example 2 Converting from Logarithmic to Exponential Form

Write each logarithmic equation in exponential form.

a. log10 1000 = 3

SOLUTION

log10 1000 = 3 10

3

= 1000 The base is the same in both the logarithmic equation and the exponential equation.

The exponential form is 103

= 1000. b. log77 = 1

SOLUTION log7 7 = 1 7

1

= 7 The logarithm is the exponent.

The exponential form is 71 = 7. c. log3 81 = x

SOLUTION log3 81 = x 3

x

= 81 The logarithm (and the exponent) can be a variable.

The exponential form is 3x

= 81.

A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log10 5.

Example 3 Application: Acidity of Rainwater

Because of the phenomenon of acid rain, the acidity of rainwater is important to environmental scientists. The acidity of a liquid is measured in pH, given by the function pH = -log[H+_]_{, where}_[_H+_]_{represents the}

concentration of hydrogen ions in moles per liter. In 1999, the hydrogen ion concentration of rainwater in western Nevada was found to be

approximately 0.0000032 moles per liter. What was the pH of the rainwater? SOLUTION

pH = -log[H+_]

pH = -log(0.0000032) Substitute the known value in the function.

Use a calculator to find the value of the logarithm in base 10. Press the key.

Reading Math The equations in Example 1d can be written 5-2₌_{_}1 25 and log5 1 _ ₂₅ = -2.

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Practice

Lesson 64 459

Check

-log (0.0000032) ≈ 5.5 Write the result found in the Solution.

log (0.0000032) ≈ -5.5 Multiply both sides by –1.

10-5.5

≈ 0.0000032 Write the logarithmic equation in exponential form.

Use a calculator to verify that the value of 10-5.5

is approximately 0.0000032.

Lesson Practice

Write each exponential equation in logarithmic form.

a. 32₌₉ _{b. 4}1₌₄

c. 90₌₁ _{d. 8}-1 ₌_0.125

Write each logarithmic equation in exponential form.

e. log10100 = 2 f. log88 = 1 g. log5125 = x

h. The acidity of a liquid is measured in pH, given by the function pH = -log[H+_]_{, where}_[_H+_]_{represents the concentration of hydrogen ions in}

moles per liter. In 1999, the hydrogen ion concentration of rainwater in northern Maine was found to be approximately 0.0000200 moles per liter. What was the pH of the rainwater, to the nearest tenth?

(Ex 1) (Ex 1) (Ex 2) (Ex 2) (Ex 3) (Ex 3)

Use synthetic substitution.

1. Find f(6) for f(x)= x4_-_4x3₊_2x2₊_4x_-_18.

2. Find f(-7) for f(x)= x4 + 5x3

- 12x2

- 4x + 8.

3. Find the equation of the line that passes through (-2, 5) and (-6, -3).

4. The sides of a triangle measure 7 in., 6 in., and 8 in. Is the triangle a right triangle? Find the zeros of each quadratic function.

5. f(x) = 2x2₊_x_-₁₅ _{6. f}₍_x_{) =}_-_32x2_-_28x _{7. f}₍_x_{) =}_x2_-₉

Determine if (0, -3) is a solution of the inequalities.

8. 5x + 2y < -4 9. y - 2x > 6 10. 16y - 2 ≥ x

11. Formulate A parasail is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasail is 48 degrees. Write a formula that would help you to estimate the parasail’s height above the boat.

(51) (51) (51) (51) (26) (26) (41) (41) (35) (35) (35)(35) (35)(35) (39) (39) (39)(39) (39)(39) (52) (52) Hint The display 3.16227766E - 6 means 3.16227766 × 10-6_, which is approximately 3.2 × 10-6_{, or 0.0000032.}

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*12. Coordinate Geometry Find the area of triangle OQR. Give the answer in exact form and to the nearest tenth of a square unit.

13. Find an equation for the inverse of y = 5x - 8. Identify the domain and range of the inverse function.

*14. Write Explain why log0 9 and log1 9 do not exist.

15. Population Currently the U.S. population is growing at a rate of approximately 0.6 percent per year. According to the 2000 census, the population was about 281 million. Use composite functions to predict the U.S. population in the year 2015, and justify your prediction.

*16. Multi-Step Use

(

_

9 7

)

1 _ ₄

(

_

9 7

)

_ 1₄ .

a. Which property of rational exponents should be applied first to simplify the expression? Explain.

b. Apply the property from part a.

c. Which property of rational exponents should be applied next? d. Apply the property from part c.

e. If possible, finish simplifying the expression from part d.

*17. Transportation The approximate volume of a moving truck, in cubic feet, that can move four to five household rooms can be given by x3_-_15x2₊_56x₊_6x2_-_90x₊_336.

Factor this expression by grouping. Then find the dimensions when x = 15. 18. Geometry The figures below show how to create Sierpinski’s Triangle. Start with a

solid black equilateral triangle (Iteration 0); at every iteration, join the midpoints of the sides of each black triangle to get a smaller triangle, and color the interior of that triangle white.

Iteration 0 Iteration 1 Iteration 2 Iteration 3 Iteration 4

Formulate an exponential function for the number of black triangles N(x) in the xth_iteration.

19. Find the slope of the line that passes through the points A

(

_ 3x , _ 1₃

)

and

B

(

_ 1₅ , _ 5x

)

, where x is any non-zero real number.

*20. Windshield Wipers A rear windshield wiper moves through an angle of 135° on each swipe. To the nearest inch, how much greater is the length of the arc traced by the top end of the wiper blade than the length of the arc traced by the bottom end of the wiper blade?

x y O 2 2 3 Q R(3, 0) O(0, 0) P

(

_√3 2 , 1 _ 2

)

x y O 2 2 3 Q R(3, 0) O(0, 0) P

(

_√3 2 , 1 _ 2

)

(63) (63) (50) (50) (64) (64) (53) (53) (59) (59) (61) (61) (57) (57) (31) (31) 14 in. 9 in. Bottom end Top end 14 in. 9 in. Bottom end Top end (63) (63)

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Lesson 64 461

21. Multiple Choice Which of the following expressions is equivalent to _ 3

5 - √ 2 ? A

_

3 - 15 √ 2 21 B 15 + 3 √ 2

_

23 C 15 - 2 √ 3

_

21 D 21 - 3 √ 2

_

15 22. Multi-Step What number needs to be added to both sides of the equation to make

x2₊_12x₊₄₂₌_{0 a perfect square? Solve the equation.}

*23. Rainwater The acidity of a liquid is measured in pH, given by the function

pH = -log[H+_]_{, where}_[_H+_]_{represents the concentration of hydrogen ions in moles}

per liter. In 1999, the hydrogen ion concentration of rainwater in the Chesapeake Bay region of Maryland was found to be approximately 0.0000316 moles per liter. What was the pH of the rainwater, to the nearest tenth? Show how to check your answer by writing a logarithmic equation and its equivalent exponential equation. (Hint: You will need to use the ≈ symbol instead of the = symbol.)

*24. Write Two number cubes are tossed once. Explain why the events, one cube shows a number less than 5 and the sum of the two cubes is a multiple of 2, are dependent.

25. Justify Give an example of an event that has a probability of 0. Explain why the probability is 0.

*26. Graphing Calculator Set the mode on your graphing calculator to a + bi by pressing

the Mode key and using the arrow keys. Then find √ -1296 ÷ √ -324 .

27. Error Analysis Rizwan tried to find the value of cosθ, where θ is an angle in standard position with the point Q(-6, 8) on its terminal side. His work is shown below. What was Rizwan’s error?

r =

√

x2 + y2 r =

√

(-6)2 + 82 r = √ 100 r = 10 cosθ=

_

x_r cosθ=

_

6 10 = 0.6

*28. Multiple Choice Which equation is equivalent to log216 = x?

A 2x₌₁₆ _{B x}2₌₁₆ _{C 16}2₌_x _{D 2}16₌_x

29. Analyze Compare and contrast correlation coefficient values of -0.45 and 0.45. *30. Falling Objects The height, in feet, of an object that is falling or is projected into

the air can be described by h = -16t2₊_v

0t + h0, where h is the height in feet after

t seconds, v0 is the initial velocity of the object in feet per second, and h0 is the

initial height of the object in feet per second. A coin is tossed from the top of a 100 foot tall building, with an initial velocity of 92 feet per second. Write the equation that models the height of the coin. Find the time when the coin will reach the ground. Round your answer to the tenths place.

(44) (44) (Inv 6) (Inv 6) (64) (64) (60) (60) (55) (55) (62) (62) (56) (56) (64) (64) (45) (45) (58) (58)