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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1 61 LESSON Understanding Advanced Factoring Warm Up New Concepts 1. Vocabulary A polynomial of degree one with two terms is a.. True or False: The GCF of -x 4 y 5 - xy + is xy. (11) (). Divide 4x - x + 5 by x - 1 using synthetic substitution. (51) 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. Hint x + = x - (-), so the value of a is -. For example, x + is a factor of x + 10x + 16 and P(-) = (-) + 10(-) + 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) = x - x - 4x + 60, x - 4 SOLUTION Use synthetic division with a = Because P(4) = 0, x - 4 is a factor of P(x) = x - x - 4x b. P(x) = x + x - 4x + 6, x + SOLUTION Use synthetic division with a = Because P(-) 0, x + is not a factor of P(x) = x + x - 4x + 6. Online Connection 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 x + 7x - 15 factors into (x - )(x + 5). Therefore, P(x) factors into (x - 4)(x - )(x + 5). 46 Saxon Algebra

2 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: a + b = (a + b)(a - ab + b ) Difference of Two Cubes: a - b = (a - b)(a + ab + b ) Math Reasoning Justify Tell why x - 1 is not a difference of cubes. Example Factoring the Sum and Difference of Two Cubes Factor each expression. a. d + 15 SOLUTION The expression is a sum of two cubes, where a = d and b = 5. d + 15 (d + 5)(d - 5d + 5) b. 7x 6 + y 9 SOLUTION The expression is a sum of two cubes, where a = x and b = y. 7x 6 + y 9 = (x ) + (y ) (x + y )(9x 4 - x y + y 6 ) c. m 5-4m SOLUTION First factor out the GCF, m. Then factor the difference of the two cubes, where a = m and b =. m 5-4m m (m - 8) m (m - )(m + m + 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 Factor each expression. a. x + 5x + x + 15 Factoring by Grouping SOLUTION The first two terms have a common factor as do the last two terms. (x + 5x ) + (x + 15) x (x + 5) + (x + 5) Factor x from the first group and from the second. (x + 5)(x + ) Factor out the GCF of (x + 5). Neither factor can be factored further. Lesson 61 47

3 Math Reasoning Analyze Can a polynomial with five terms be factored by using the grouping method? Explain. b. x 4 + 4x - x - 4 SOLUTION Factor x from the first two terms and -1 from the last two terms. x 4 + 4x - x - 4 x (x + 4) - 1(x + 4) (x + 4)(x - 1) Factor out the GCF of (x + 4). (x + 4)(x - 1)(x + x + 1) Factor the difference of cubes. c. x 6 + x 5 + x 4 + x + x + SOLUTION Use two groups of three terms each. (x 6 + x 5 + x 4 ) + (x + x + ) x 4 (x + x + ) + 1(x + x + ) (x + x + )(x 4 + 1) Factor out the GCF of (x + x + ). (x + 1)(x + )(x 4 + 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 x - x + 170x 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 The expression for the volume factors into (x - 5)(x - 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) = x + 19x + x + 16, x + (Ex 1) (Ex 1) b. P(x) = 4x - x + 56x - 1, x - 6 Factor each expression. c h d. s 6 t 1-15r (Ex ) (Ex ) e. 8 7 x7 + x 4 f. xy - 9y + 5x - 45 (Ex ) (Ex ) (Ex ) g. 4m n + 1m - n - h. x 5 + 1x 4 + 6x + 5x + 60x (Ex ) (Ex ) (Ex 4) i. The volume of a rectangular solid with a length of x - 15 feet has a volume of x + 5x - 600x cubic feet. Factor the expression for the volume completely. 48 Saxon Algebra

4 Practice Distributed and Integrated Graph. 1. x = -5. x - y = 4. 7y + 7x = 1 (1) (1) (1) (Inv ) 4. Find the intercepts of the equation 5x - 4y + 7z = 1. Simplify (59) (59) (59) Find an inverse for the equations. 8. y = x - 9. y = 5x + 15 (50) (50) 10. Let K(u) = 17 and L(u) = -1. Find K (L(u)) and L (K(u)). (5) *11. Minimize P = 4x + y for the constraints (54) x 4 y 6 x + y 7 *1. Mary deposited $980 at 7.0% compounded continuously. How much money will (57) she have after 9 years? 1. Meteorology The number of tornadoes in the United States in 006 is shown in (55) the table. Season Jan-Feb Mar-Apr May-Jun Jul-Aug Sep-Oct Nov-Dec Number of Tornadoes 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. (61) 15. Probability A student graphed a set of data points and found that the correlation (45) coefficient was What is the probability that the slope of the line of best fit is greater than 1? Explain. *16. Geometry The probability that a randomly selected point lies within a (60) 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. 10 in. 6 in. in. Lesson 61 49

5 17. Buildings One building is 10 feet tall and a nearby building is 74 feet tall. From (46) the top of the taller building, you can spot the top of the shorter building at an angle of depression of 0. 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 (5) your shadow changes as you approach each lamppost. The length of the shadow seems to be 1 5 of your distance to the lamppost. Your dog is dragging you along at about 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 (59) give the correct solution. x 8 = ( x 8 ) 8 = () 8 x 1 = 8 x = Analyze Suppose you spin the spinner shown 8 times. What is the probability (49) that you will spin black 5 times? 1. Verify Use the form x + cx + c for a quadratic perfect square trinomial to verify (58) the method of completing the square. *. Graphing Calculator Describe the feasible region for the inequalities: (54) x 0, y 0, y -0.75x + 8. Multiple Choice Which value of r, the correlation coefficient, describes the set of (45) points that are closest to forming a line? A r = B r = 0 C r = 0.6 D r = 0.65 *4. Commuting to Work According to data collected in the 005 American Community (60) Survey, approximately 10.67% of workers carpool to work, 4.66% use public transportation, and.47% walk. Find the probability that a randomly selected worker carpools or uses public transportation. 5. Write A few different equations of the form y = b x are graphed to the (47) right. Describe in your own words how the shape of the graph depends on the value of b. y =.5 x O y y =.5 x y = 1.5 x x Saxon Algebra

6 *6. Multi-Step Use (59) 5 5 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. 7. Multiple Choice In the diagram to the right, find the length of the (5) hypotenuse of triangle LMN. A 7.5 B C 15 D 15 N L 15 cm M *8. Multi-Step The area of a twin size mattress is x x + 5x square inches. (61) a. Factor the polynomial by grouping. b. Find the area given that x = Amusement Parks The Giant Swing at Silver Dollar City in Branson, Missouri, rotates (56) through an angle measuring 0. Find the reference angle for this angle. *0. Verify Show that there are two different ways to group the terms in (61) d + 6f + d + df and that both lead to the same factorization. Lesson

7 6 LESSON Using Complex Numbers Warm Up New Concepts Math Reasoning Generalize For any positive real n, -n =. 1. Vocabulary Any point on the number line is a number. (1). Simplify (5 + x) - (6 - x). (). Solve 5x - 15 = 0. (58) 4. Simplify 150. (40) 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 x = -1. Because i = -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 SOLUTION Write -49 as the product of 49 and The square root of a product is the product of the square roots. - 7 i 49 = 7 and -1 = i. -14i Multiply the real numbers. b. Simplify Online Connection SOLUTION Write -117 as a product with a perfect square The square root of a product is the product of the square roots. 1 (i) 9 = and -1 = i. i 1 Commutative Property of Multiplication 44 Saxon Algebra

8 Example Solving a Quadratic Equation with Imaginary Numbers Solve x + 5 = 0. Write the solutions in terms of i. Check the answers. SOLUTION x + 5 = 0 x = -5 Subtract 5 from each side. x =± -5 Take the square root of each side. x =±5i Simplify the square root. Math Language The solutions are the conjugate pairs 5i and -5i. Complex solutions of quadratic equations always occur in conjugate pairs of the form a + bi and a - bi. Check x + 5 = 0 (5i) i (-1) = 0 x + 5 = 0 (-5i) i (-1) = 0 An imaginary number bi is part of a complex number. Math Reasoning Analyze Are real numbers a subset of complex numbers or are complex numbers a subset of real numbers? 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 + i + 0i = 0 + i = i 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 Simplifying Expressions with Complex Numbers Add or subtract. Write the answer in the form a + bi. a. (7 + 4i) + (- - 5i) SOLUTION (7 + (-)) + (4 + (-5))i = 5 + (-1)i = 5 - i b. (5 - i) - (5-11i) SOLUTION (5-5) + (- - (-11))i = 0 + 8i = 8i Lesson 6 44

9 Example 4 Solving Quadratic Equations Solve each equation. Write the solutions in the form a + bi. a. (x - ) = -4 SOLUTION (x - ) = -4 x - = ± -4 Take the square root of each side. x - = ±i Simplify the square root. x - = i or x - = -i Solve both equations. x = + i or x = - i The solutions are + i and - i. b. x + 8x + 18 = 0 SOLUTION x + 8x = -18 x + 8x + 16 = Subtract 18 from each side. Complete the square. (x + 4) = - Write the left side as a perfect square. x + 4 = ± - Take the square root of each side. x + 4 = ±i Simplify the square root. x = -4 ± i Subtract 4 from each side. The solutions are -4 + i and -4 - i. Lesson Practice a. Simplify (Ex 1) b. Simplify (Ex 1) (Ex ) (Ex ) c. Solve 0 = x. Write the solutions in terms of i. Check the answer. d. Solve -6x = 16. Write the solutions in terms of i. Add or subtract. Write the answer in the form a + bi. (Ex ) e. (- + i) + (14 + 4i) f. (10-6i) - (-4 + i) Solve each equation. Write the solutions in the form a + bi. (Ex 4) g. (x + 9) = -9 h. x - x + = 0 Practice Distributed and Integrated Factor. *1. 15x x - x - *. 64x 1-15y 9 (61) (61) 444 Saxon Algebra

10 Add or subtract.. a + a (7) b 4. ax m p - c + m (7) (7) (7) gh 5. - x + xh + g h gx Factor the greatest common factor. 6. 5x 7 y 5 m - 7x 5 m y + 14y 7 x 4 m 7. 6x ym 5 - x ym + 4xym () () Simplify (-7) (59) (59) 81 4 (59) (59) School It is estimated that 15% of students walk to school. What is the probability (49) that exactly out of 5 randomly selected students will walk to school? 1. Find an equation for the inverse of y = 1 5 x + 6. (50) 1. What are the odds in favor of rolling a on a number cube? (55) *14. Multiple Choice Which is a real number? (6) A 15i B i C 5 + 0i D i *15. Verify Show that the expression (7) x + 1 is equivalent to x + 1 x +. *16. Electrical Engineering A circuit has a current of (4 - i) amps, and a second circuit (6) 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. (55) 18. Projectile Motion The Leaning Tower of Pisa measures feet. If an object is (58) 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 = -16t + 80t Find the time at which the object will reach a height of 00 feet. Round your answer to the tenths place. *19. A number cube is rolled once. Are the events a number greater than or a multiple (60) of mutually exclusive or inclusive? Explain why. *0. Transportation The approximate volume of a moving truck, in cubic feet, can be (61) given by x - 6x + 8x + 5x - 0x Factor this expression by grouping. Then find the volume when x = Multiple Choice Which of the following angles is not coterminal with 5º? (56) A 585º B -15º C 95º D -495º. Multi-Step The population of an insect colony is growing at a rate of 1.5% per week. (57) 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? Lesson 6 445

11 *. Graphing Calculator Factor x 5 - x 4 - x + 1. Then enter the original expression for (61) Y1 and the factored expression for Y on a graphing calculator. Access the Table function and study the table of values for the expressions. What is true? What does it mean? 4. Write Explain why radical expressions with even indices and positive radicands (59) have two real roots. Use 6 64 to demonstrate your explanation. *5. Geometry An expression for the volume of the rectangular (51) prism is V = x + 7x - 14x Find an expression for the missing dimension. x + 6. Analyze Suppose the graph of y = b x + k passes through the (57) points (, -) and (5, 0). Determine whether the equation models exponential growth or decay. x + 4? 7. Error Analysis A student made an error while studying the relation (5) between the two triangles shown. Explain the error. Find the missing angles. m Y = 90-0 = 60 m R = = 0 m X = 90 m Y = 60 m Z = 0 Y X 0 Z P 60 Q m P = 90 m Q = 60 m R = 0 The corresponding angles are congruent. So, the triangles are congruent. R 8. Probability A sales clerk has a weekly quota of 15 sales. The probability, p, that the (58) sales clerk will meet this quota after working d days, in a 5-day work week, can be approximated by p = -0.04d + 0.4d. How many days a week should the sales clerk work for the probability of making 15 sales to be 0.8? *9. Verify Show that 11i and -11i are solutions of x = -11. (6) 0. Cell Phones Mobile-U has two cell phone plans, the Metropolitan Plan and the (54) Continental Plan. The cost for each plan is shown in the table. A company is comparing the two plans. The company needs at least 1 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 Per minute rate above free 10 cents 7 cents minutes Monthly charge $0 # of cell phones $5 # of cell phones 446 Saxon Algebra

12 6 LESSON Understanding the Unit Circle and Radian Measures Warm Up 1. Vocabulary Half the diameter of a circle is the of the circle. (SB). If both legs of a right triangle have length 1, then the length of the hypotenuse is. (41). If the shortest leg of a triangle has length 1, then the lengths of the other two sides of the triangle are and. (5) New Concepts A unit circle is a circle with a radius of 1 unit. 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 triangle? How do you know?. Use what you know about special right triangles to find the lengths of the legs of the triangle.. 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 y y x x 1 1 P(x, y) y y x Q(x, y) x Reading Math Remember the symbol θ is read theta. Online Connection 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. r θ = 1 radian r Lesson 6 447

13 There are 60 or π radians in a circle. Thus the conversion factor π 60, which is equivalent to 1, can be used to convert measures θ = -10 = - 7π 6 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 -10 angle is equivalent to an angle measure of 150. y θ = 60 = π radians x Converting Angle Measures Degrees to Radians Radians to Degrees Multiply the number of degrees Multiply the number of radians by ( π radians 180 ). by 180 ( π radians ). Math Language Angle measures in radians often appear without the label radians. For example, - π 6 radian is often written simply as - π 6. Example 1 Converting Between Degrees and Radians Convert each measure from degrees to radians or from radians to degrees. a. -0 SOLUTION -0 ( π radians 180 ) = - π b. π 4 radians SOLUTION ( π 6 radians Multiply by ( π radians 180 ). 4 radians 180 ) ( π radians ) = 15 Multiply by ( 180 π radians ). For every point P(x, y) on the unit circle, the value of r is 1. Therefore, if θ is an angle in standard position whose terminal side passes through point (x, y) on the unit circle, then: sin θ = y r = y 1 = y cos θ = x r = x 1 = x tan θ = y x So, the coordinates of P can be written as (cos θ, sin θ). The Unit Circle ( 1 (0, 1) ( 1 π,, ) ) ( π π, ) 90 ( π π, ) ( π 5π ( 6, 1 ), 1 ) (-1, 0) π 180 II I 0 0 (1, 0) III IV π 11π π 5π 40 ( π 4, 1 (, 1 ) 4π π (, ( ) ), (0, -1) ( 1, ( 1 ), ) The diagram shows the equivalent degree and radian measures of special angles, and the corresponding x- and y-coordinates of points on the unit circle. ) 448 Saxon Algebra

14 Example Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find each trigonometric function value. Find exact values. a. sin 10 Hint The exact value of sin 10 is irrational. An approximate value is sin 10 = exact approximate SOLUTION The terminal side of 10 passes through the point (- 1, ) on the unit circle. sin θ = y sin 10 = b. tan 11π 6 SOLUTION The terminal side of 11π passes through the point 6 (, - 1 ) on the unit circle. tan θ = y x tan 1-11π 6 = = - 1 = - 1 = - 1 = - To divide by a fraction, multiply by its reciprocal. Rationalize the denominator. 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 : Use the portion of the unit circle in Quadrant I to find the sine, cosine, or tangent of the reference angle. Step : Use the quadrant of the terminal side of θ in standard position to determine the sign of the sine, cosine, or tangent. Lesson 6 449

15 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. QII QIII sinθ + sinθ + cosθ cosθ + tanθ tanθ + sinθ sinθ cosθ cosθ + tanθ + tanθ QI QIV Hint Recall that a reference angle is an acute angle. So for θ < 60, the measure of θ is: QII θ QIII - θ QIV 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 Example Using Reference Angles to Evaluate Trigonometric Functions Find the sine, cosine, and tangent of 15. Find exact values. SOLUTION Step 1: Find the measure of the reference angle. The measure of the reference angle is 45. Step : Find the sine, cosine, and tangent of the reference angle. sin 45 = Use sin θ = y. cos 45 = Use cos θ = x. tan 45 = 1 Use tan θ = y x. Step : Determine the sign. sin 15 = In Quadrant II, sin θ is positive. cos 15 = - tan 15 = -1 In Quadrant II, cos θ is negative. In Quadrant II, tan θ 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 45 y 45 y 15 x x (, ) θ r s 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 = θ π πr or s = θr. 450 Saxon Algebra

16 Example 4 Finding Arc Lengths a. Find the length of arc s 1. Approximate to the nearest tenth. SOLUTION s = rθ s 1 = 5 π = 10π 10.5 cm Write the formula. Substitute s 1 for s, 5 for r, and π Simplify. Use a calculator to approximate. for θ. π s 1 5 cm b. Find the length of arc s. Approximate to the nearest tenth. SOLUTION Step 1: Convert 5 to radians. 5 ( π radians 180 ) = 5π 4 radians Step : Use the formula for arc length. s 5.5 ft s = rθ Write the formula. s =.5 ( 5π 4 ) Substitue s for s,.5 for r, and 5π 4 for θ. =.15π Simplify. 9.8 ft Use a calculator to approximate. Example 5 Application: Planet Rotation Earth makes one complete rotation in 4 hours, and its radius is approximately 955 miles. If an object is fixed on Earth s equator, how far does it travel in 1 hour due to Earth s rotation? 1 4 of a rotation in 1 hour SOLUTION r = 955 mi The central angle that corresponds to a complete rotation is π radians. In 1 hour, Earth makes 1 of a complete rotation. So, θ = 1 π = π s = rθ = 955 π The object travels about 105 miles in 1 hour. Check Find the approximate circumference of Earth: πr π 955 4,850 miles. 1 of 4,850 is slightly greater than 1000, so 105 miles is reasonable. 4 Lesson 6 451

17 Lesson Practice a. Convert 150 to radians. b. Convert - 4π radians to degrees. (Ex 1) (Ex 1) c. Use the unit circle to find the exact value of cos 15. d. Use the unit circle to find the exact value of tan π (Ex ) (Ex ) e. Use a reference angle to find the sine, cosine, and tangent of 10. Find exact values. (Ex ) (Ex 4) 4. f. Find the length of arc s 1. Approximate to the nearest tenth. s 1 4π 8.5 m g. Find the length of arc s. Approximate to the nearest tenth. (Ex 4) 15 8 in. s h. The minute hand of a certain clock is 15 centimeters long. How far does the tip of the minute hand travel in 5 minutes, to the nearest centimeter? (Ex 5) Practice Distributed and Integrated Simplify x + y 4 + xy 1. 4 x - 7 (48) (48) (48). x + (48) x + 1 x x - x Determine if the following probability experiments are binomial experiments. If not, explain why. 4. Rolling a number cube 18 times and recording the results (Inv 5) (Inv 5) 5. Flipping coins 0 times to test if they match half of the time Simplify (4 - ) (40) (40) Expand. 8. (x + 5) 9. (x + 4) (19) (19) 45 Saxon Algebra

18 Solve x - 6 = x = -11x - x (5) (5) 1. Error Analysis Julian is finding the inverse of the function y = x. Which step is (50) incorrect? Step One: y = x Step Two: x = y Step Three: x = y Step Four: ± x = y *1. Write the expression in terms of i and then add: ( ) + ( ). (6) 14. Geometry In the triangle to the right, find x. (44) x m C 90 x m B 45 9 m 45 A *15. Planet Rotation Jupiter is the largest planet. Its radius is approximately 44,65 (6) miles. Jupiter rotates faster than Earth. It takes approximately 9.8 hours for Jupiter to make one complete rotation. If an object is fixed on Jupiter s equator, how far does it travel in 1 hour due to Jupiter s rotation? 16. Multiple Choice Which of the following functions could be used to model (57) exponential growth? A y = 1 B y = e -x e x C y = e x D y =.4 e -x 17. Surveying A surveyor stands on the roof of a building that is 180 feet above the (46) ground and spots the top of a taller building at an angle of elevation of 4. The two buildings are 480 feet apart. Find the height of the taller building. Round to the nearest foot. *18. Graphing Calculator Describe the feasible region for the set of inequalities x 0, (54) y 0, y -1.5x *19. Formulate A fair coin is flipped. Write the equation that can be used to solve for (60) 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. 1 x + 1 y 0. Multiple Choice Which of the following expressions is equivalent to (48) 1-1 A 1 - x B x - xy - x - y C x + y (1 - x) x + y x y xy - y D x + y *1. Write How is adding and subtracting complex numbers similar to adding and (6) subtracting polynomials? *. Model Sketch a graph of five points whose correlation coefficient (45) is about 0.1. x? Lesson 6 45

19 180 (n - ) n. Multi-Step The formula θ = gives the measure of each interior angle of (56) 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. 4. Estimate Use what you know about the unit circle and special angles to estimate (6) cos(0.π). Do not use a calculator. Explain your method. 5. Jupiter Gravity on the planet Jupiter is almost three times stronger than on Earth. (58) 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 = -4t + 100t. Find the time at which the object will first reach 50 feet. Round your answer to the tenths place. 6. Analyze Let f(x) = ax + b and g(x) = cx + d. Under which conditions will the (5) composite functions f(g(x)) and g(f(x)) be equal? 7. Construction The volume of a typical American-made brick in cubic millimeters (61) can be represented by the polynomial x - 0x x + 16,000 where the length is given by x Fully factor the expression for the volume of a brick. 8. Coordinate Geometry Sketch a graph of the equation y = x, where y is the side (59) 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.4 in. Use the graph to estimate the side length of a cube with a volume of 6.9 in. *9. Error Analysis A student incorrectly found s, the length of the (6) indicated arc, as shown below. s = rθ = = 600 m What is the error? Find the correct arc length m s *0. Pass codes A computer generates temporary pass codes for users. Each pass code (55) is exactly letters long with no repeating letters. What is the probability of getting a pass code with vowels (A, E, I, O, or U)? 454 Saxon Algebra

20 LAB 10 Using the Log Keys Graphing Calculator Lab (Use with Lesson # 64, 7, 81, 87, 9, 10, and 110) 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. Calculate the natural logarithmic expression 4 ln (5). a. To enter the value 4 ln (5), press. b. To calculate this value, press Hint Use the Change of Base Law when calculating a log that does not have a base of 10 or e... 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 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. 1. Graph the common logarithmic function log (x + ). a. Press to access the screen to enter the logarithmic function. b. To enter the function log (x + ), press., and then press c. Press the ZStandard window. d. Then press to select to view the graph. Online Connection Lab

21 . Graph the logarithmic function log6 (1x) whose base is not natural or common. a. Press to access the screen to enter the logarithmic function. b. To enter the function log6 (1x), press., and then press c. Press the ZStandard window. d. Then press to view the graph. Lab Practice 1. Calculate 6 log (8).. Calculate ln (1).. Calculate log7 (9). 4. Graph y = log (15x + 8). 5. Graph y = log5 (x + 6). 456 Saxon Algebra to select

22 64 LESSON Using Logarithms Warm Up New Concepts 1. Vocabulary In the expression 4 5, the base is 4 and the is 5. (). What is the value of? (). What value of x makes the equation 10 x = 100 true? (47) 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 log b 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. 5 = Reading Math Read the symbol as if and only if. When used between two equations, it means the equations are equivalent. SOLUTION 5 = log = 5 The base is the same in both the exponential equation and the logarithmic equation. The logarithmic form is log = 5. b. 5 1 = 5 SOLUTION 5 1 = 5 log 5 5 = 1 The exponent is the logarithm. The logarithmic form is log 5 5 = 1. c. 8 0 = 1 SOLUTION Online Connection = 1 log 8 1 = 0 Any nonzero base to the zero power is 1. The logarithmic form is log 8 1 = 0. Lesson

23 Reading Math The equations in Example 1d can be written 5 - = 1 5 and log = -. d. 5 - = 0.04 SOLUTION 5 - = 0.04 log = - An exponent (or log) can be negative. The logarithmic form is log = -. Example Converting from Logarithmic to Exponential Form Write each logarithmic equation in exponential form. a. log = SOLUTION log = 10 = 1000 The exponential form is 10 = b. log 7 7 = 1 SOLUTION The base is the same in both the logarithmic equation and the exponential equation. log 7 7 = = 7 The exponential form is 7 1 = 7. The logarithm is the exponent. c. log 81 = x SOLUTION log 81 = x x = 81 The exponential form is x = 81. The logarithm (and the exponent) can be a variable. 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 = log Example 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 moles per liter. What was the ph of the rainwater? SOLUTION ph = -log[h + ] ph = -log( ) Substitute the known value in the function. Use a calculator to find the value of the logarithm in base 10. Press the key. The rainwater had a ph of about Saxon Algebra

24 Hint The display e - 6 means , which is approximately. 10-6, or Check -log ( ) 5.5 Write the result found in the Solution. log ( ) -5.5 Multiply both sides by Write the logarithmic equation in exponential form. Use a calculator to verify that the value of is approximately Lesson Practice Write each exponential equation in logarithmic form. (Ex 1) a. = 9 b. 4 1 = 4 c. 9 0 = 1 d. 8-1 = 0.15 Write each logarithmic equation in exponential form. (Ex ) e. log = f. log 8 8 = 1 g. log 5 15 = 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 moles per liter. What was the ph of the rainwater, to the nearest tenth? (Ex ) Practice Distributed and Integrated Use synthetic substitution. 1. Find f(6) for f(x) = x 4-4x + x + 4x (51). Find f(-7) for f(x) = x 4 + 5x - 1x - 4x + 8. (51). Find the equation of the line that passes through (-, 5) and (-6, -). (6) 4. The sides of a triangle measure 7 in., 6 in., and 8 in. Is the triangle a right triangle? (41) Find the zeros of each quadratic function. 5. f(x) = x + x f(x) = -x - 8x 7. f(x) = x - 9 (5) (5) (5) Determine if (0, -) is a solution of the inequalities. 8. 5x + y < y - x > y - x (9) (9) (9) 11. Formulate A parasail is attached to a boat with a rope 00 feet long. The angle of (5) 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. Lesson

25 *1. Coordinate Geometry Find the area of triangle OQR. Give the answer in (6) exact form and to the nearest tenth of a square unit. O(0, 0) y P(, 1 ) Q O x R(, 0) 1. Find an equation for the inverse of y = 5x - 8. Identify the domain and range of the (50) inverse function. *14. Write Explain why log 0 9 and log 1 9 do not exist. (64) 15. Population Currently the U.S. population is growing at a rate of approximately (5) 0.6 percent per year. According to the 000 census, the population was about 81 million. Use composite functions to predict the U.S. population in the year 015, and justify your prediction ) ( ). (59) a. Which property of rational exponents should be applied first to simplify the expression? Explain. b. Apply the property from part a. *16. Multi-Step Use ( 9 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 (61) move four to five household rooms can be given by x - 15x + 56x + 6x - 90x + 6. Factor this expression by grouping. Then find the dimensions when x = Geometry The figures below show how to create Sierpinski s Triangle. Start with a (57) 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 Iteration Iteration 4 Formulate an exponential function for the number of black triangles N(x) in the x th iteration. 19. Find the slope of the line that passes through the points A ( x, 1 ) and (1) B ( 1 5, 5 x ), where x is any non-zero real number. *0. Windshield Wipers A rear windshield wiper moves through (6) an angle of 15 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? Top end Bottom end 14 in. 9 in. 460 Saxon Algebra

26 1. Multiple Choice Which of the following expressions is equivalent to (44)? A B C D Multi-Step What number needs to be added to both sides of the equation to make x + 1x + 4 = 0 a perfect square? Solve the equation. (Inv 6) *. Rainwater The acidity of a liquid is measured in ph, given by the function (64) 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 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.) *4. Write Two number cubes are tossed once. Explain why the events, one cube (60) shows a number less than 5 and the sum of the two cubes is a multiple of, are dependent. 5. Justify Give an example of an event that has a probability of 0. Explain why the (55) probability is 0. *6. Graphing Calculator Set the mode on your graphing calculator to a + bi by pressing (6) the Mode key and using the arrow keys. Then find Error Analysis Rizwan tried to find the value of cosθ, where θ is an angle in standard (56) position with the point Q(-6, 8) on its terminal side. His work is shown below. What was Rizwan s error? r = x + y r = (-6) + 8 r = 100 r = 10 cosθ = x r cosθ = 6 10 = 0.6 *8. Multiple Choice Which equation is equivalent to log 16 = x? (64) A x = 16 B x = 16 C 16 = x D 16 = x 9. Analyze Compare and contrast correlation coefficient values of and (45) *0. Falling Objects The height, in feet, of an object that is falling or is projected into (58) the air can be described by h = -16t + v 0 t + h 0, where h is the height in feet after t seconds, v 0 is the initial velocity of the object in feet per second, and h 0 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 9 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. Lesson

27 65 LESSON Using the Quadratic Formula Warm Up New Concepts Math Reasoning Analyze Explain how -c a + b 4a 4ac becomes - 4a + b 4a. 1. Vocabulary The square root of a negative number is an number.. Solve by completing the square x - 6x = 1. (6) (58). Simplify -5. (6) 4. Evaluate b - 4ac when a =, b = 7, and c = 5. () In general, a quadratic equation in standard form is ax + bx + c = 0, with a 0. The general equation can be solved for x by completing the square. x + b ax + bx = -c x + b a x = - c a a x + b 4a = - c ( x + b a ) = - c ( x + b 4ac ( x + b a + b 4a a + b 4a a ) = - 4a + b 4a a ) = b - 4ac 4a x + b a = ± b - 4ac 4a b x = - a ± b - 4ac Isolate x. 4a b x = - a ± b - 4ac a x = -b ± b - 4ac a The last line is the quadratic formula. Subtract c from each side. Divide both sides by a. Complete the square. Factor the left side. Make common denominators on the right. Combine the fractions on the right. Apply the Square Root Property. Simplify the radical. Combine the fractions. Quadratic Formula The solutions of the quadratic equation ax + bx + c = 0 (a 0) are given by x = -b ± b - 4ac. a Online Connection The quadratic formula can be used to find the solutions of any quadratic equation. 46 Saxon Algebra

28 Example 1 Solve each equation. Solving Quadratic Equations with Real Zeros Math Language The solutions of a quadratic equation are the zeros of the related function. a. 5x + 4x - 7 = 0 SOLUTION Use the quadratic formula. x = -b ± b - 4ac a = -4 ± 4-4(5)(-7) (5) 196 = -4 ± 10-4 ± 6 = 10 The solutions are 1 5 and -7. Substitute 5 for a, 4 for b, and -7 for c. Simplify the radicand and denominator. = 1 and -7 Evaluate the square root and simplify. 5 b. 9x + 6x + 1 = 0 SOLUTION Use the quadratic formula x = -b ± b - 4ac. a x = -6 ± 6-4(9)(1) (9) 0 = -6 ± 18 = -6 ± 0 18 The solution is - 1. = = - 1 Substitute 9 for a, 6 for b, and 1 for c. Simplify. Example Solving Quadratic Equations with Complex Zeros Hint The standard form of a complex number is a + bi. a. Solve the equation x + 5x + 4 = 0. Write the solutions as complex numbers in standard form. SOLUTION Use the quadratic formula. x = -b ± b - 4ac a = -5 ± 5-4()(4) () - = -5 ± 6 = -5 ± i 6 Substitute for a, 5 for b, and 4 for c. Simplify the radicand and denominator. Write the negative root as an imaginary number. = ± i Write the solutions in standard form. 6 The solutions are i and i. 6 - Lesson 65 46

29 b. Solve the equation x - 1 x + 1 = 0. Write the solutions as complex 8 numbers in standard form. SOLUTION To make the calculations simpler, clear the fractions before using the quadratic formula. 8x - 4x + 1 = 0 Multiply both sides of the equation by 8. Use the quadratic formula: x = -b ± b - 4ac. = -(-4) ± (-4) - 4(8)(1) (8) = 4 ± = 4 ± 4i 16 = 1 ± i 4 a Substitute 8 for a, -4 for b, and 1 for c. Simplify the radicand and denominator. Write the negative root as an imaginary number. Factor 4 out of the numerator and denominator. = 1 4 ± 1 i Write the solutions in standard form. 4 The solutions are i and i. Because the solutions of a quadratic equation are the zeros, or x-intercepts of the equation, a graphing calculator can be used to find the solutions of a quadratic equation when the solutions are real. Math Reasoning Verify Graph y = x + 5x + 4. How does the graph confirm that there are no real solutions to x + 5x + 4 = 0? Example Solving Quadratic Equations using a Graphing Calculator Solve x + x - 15 = 0 by using a graphing calculator. SOLUTION Graph the related function, y = x + x If needed, adjust the window so that the x-intercepts of the graph are displayed. Access the Calc function (nd + Trace) and choose zero. When asked for left and right bounds, choose points that are on either side of one of the x-intercepts. Repeat for the second intercept. The solutions are - and Saxon Algebra

30 Example 4 Application: Travel Train A leaves the station at 1 p.m. and travels due north at 55 miles per hour. Train B leaves the same station at 4 p.m. and travels due east at 5 miles per hour. About how long will it take for the trains to be 500 miles apart? SOLUTION Write each distance as rate time. Train B is hours behind Train A. Train A: d = rt = 55t Train B: d = rt = 5(t - ) 55t 500 a + b = c (55t) + (5(t - )) = t + (5t - 105) = 50,000 05t + 15t - 750t + 11,05 = 50, t - 750t - 8,975 = 0 Now use the quadratic formula. t = -(-750) ± (-750) - 4(450)(-8,975) (450) = 750 ± 4,116,597, t 8.41 and t Use the Pythagorean Theorem. Simplify. Square the binomial. Combine like terms. Disregard the negative time. It will take about 8.41 hours for the trains to be 500 miles apart. Check Find the distance each train traveled after 8.41 hours. Train A: 55(8.41) = miles, Train B: 5( ) = miles Use the Pythagorean Theorem: = (t - ) Lesson Practice Solve each equation. (Ex 1) a. x - 7x + = 0 b. 9x + 1x + 4 = 0 Solve each equation. Write the solutions as complex numbers in standard form. c. x + 5x = - 9 (Ex ) d. -x + 1 x = 0 (Ex ) (Ex ) (Ex 4) e. Solve 8x + 7x = 0 by using a graphing calculator. f. A hiker leaves a campground at 8 a.m. and heads due south at a rate of 4 miles per hour. Another hiker leaves the same campground at 9 a.m. and heads due east at a rate of miles per hour. Approximately how long will it take for the hikers to be 15 miles apart? Lesson

31 Practice Distributed and Integrated 1. Secret ingredient X has a half-life of hours. What portion of 60 kg will remain after 7 hours? (57) Use long division.. Divide (x + 1x - 15x + 15) by (x - 9) (8). Divide (x + 14x - 4x - 48) by (x + 4) (8) Solve. *4. x = 7 + x *5. x = -x + 1 (58) (58) Simplify. 6. (5-6 ) (40) (40) (40) 9. Jordon spins a spinner with six equal-sized sections numbered 1 6. In one spin, what is the likelihood that the spinner will stop on a 1 or a 5? (55) *10. Error Analysis Find and correct the error a student made in solving 4x - 7x + 9 = 0. (65) x = -b ± b - 4ac a = -7 ± (-7) - 4(4)(9) 8 = -7 ± ± 15 = = 8-7 ± 5 = - 1 and List the coefficients in the 6 th row of Pascal s Triangle. (49) 1. Probability A history test has 0 questions, 7 of which are about the Civil War. (55) Students are asked to answer 5 of the 0 questions. What is the probability that a student randomly selecting questions will select 5 Civil War questions? *1. Write Describe how to determine sin π (6) by using the unit circle. 14. Find an equation of a line perpendicular to y = 1 x - 4. (6) 15. Analyze In what type of right triangles will the sine and cosine values of the acute (46) angles be equal? Find x and y (5) (5) y x 5 y 0 x Saxon Algebra

32 *18. Multi-Step Complete parts a d to graph a logarithmic function and use the graph (64) to estimate a logarithm. a. Write the equation y = log 1 4 in exponential form. What value of y makes both equations true? b. Writing exponential equations if needed, complete the table of values for the function y = log x. x y = log x c. Plot the points and sketch the graph of the function. d. Use your graph to estimate the value of log 6. Check your estimate by evaluating the appropriate power of on a calculator. 19. Multiple Choice Choose the letter that best represents the value of x in the equation (58) x + x + 7 = 8. A x = -1 ± B x = -, 0 C x =, 0 D x = 1 ± 4 0. Measurement Given that 1 strip of red ribbon = inches and 1 strip of blue ribbon (9) = 4 inches, write and graph the inequality that represents that the total number of inches must be no less than Multi-Step A car is sitting outside in a driveway. Consider the following statement, If it starts raining, the car will get wet. a. Is the converse of the statement true? Explain. b. Is the contrapositive true? Explain. (Inv 1). Geometry The base of a triangle is 5 feet, and its height is 1 (44) area of the triangle. feet. Find the *. Physics A fireworks technician wants a firework to explode 70 feet below the (6) top of the Gateway Arch in St. Louis. The Gateway Arch has a height of 60 feet. The technician has figured the height of the firework to be modeled by -16t + 160t, where t is the time in seconds after it is launched. The solutions of -16t + 160t = 560 give the number of seconds it will take for the firework to reach the desired height. Solve the equation. Will the firework reach the desired height? If so, when? 4. Justify Let f(x) = x - 7 and g(x) = 1 x + 7 (5) a. Justify the claim that f (g(x)) = x. b. Does g(f(x)) = x also? Explain. 5. Error Analysis Find and correct the error a student made in factoring the expression (61) below.. 9x 4-6x - 8x x (x - 7) - 8(x - 7) (x - 7)(9x - 8) (x - 7)(x - )(6x + 6x + 4) Lesson

33 *6. Physical Science The North Falls waterfalls in Silver Falls State Park in Oregon are (65) 16 feet high. The equation -16t + 16 = 0 gives the number of seconds it takes for a drop of water at the top of the falls to reach the pool of water below. Solve the equation by using the quadratic formula. Round to the nearest hundredth of a second. 7. Formulate Let θ be an angle in standard position and let n be any integer. You can (56) use the expression θ + n(60 ) to indicate all the angles coterminal with θ. Write an expression for all the angles coterminal with 15, and find different ones. *8. Graphing Calculator Solve 5x - x - 16 = 0 by using a graphing calculator. (65) 9. Velocity of an Electron The velocity at which an electron orbits the nucleus of (59) an atom is known as the Fermi-Thomas velocity and can be found using the equation Z 17 c, where Z is the number of protons in the nucleus and c is the speed of light. What is the Fermi-Thomas velocity for platinum, which has 78 protons, in terms of c? Round to the nearest thousandths place. *0. Analyze Write the following equations in exponential form. What property of (64) exponents is illustrated? log 1 = 0, log 1 = 0, log 15 1 = Saxon Algebra

34 66 LESSON Solving Polynomial Equations Warm Up New Concepts Hint Recall that the Zero Product Property states that if ab = 0, then a = 0 or b = Vocabulary The formula x = -b ± b - 4ac is called the. a. Factor 7cd - 1cd. (65) (). Solve x(x - 4) = 0. (5) 4. True or False: x + 7 is a factor of x + 8x + x - 4. (61) 5. Solve x + x + 1 = 0 by using the quadratic formula. (65) Sometimes a polynomial equation can be solved just by factoring and using the Zero Product Property. Example 1 Solve each equation. a. x 4-10x - 7x = 0 Using Factoring to Solve Polynomial Equations SOLUTION Since one side is already equal to 0, begin by factoring the polynomial. x (x - 5x - 6) = 0 The GCF is x. x (x + 4)(x - 9) = 0 x = 0 or x + 4 = 0 or x - 9 = 0 Factor the trinomial. Use the Zero Product Property. x = 0 or x = -4 or x = 9 Solve each equation. The solutions are 0, -4, and 9. b. 4x + x = 4x SOLUTION First write the equation in standard form. 4x - 4x + x = 0 x(4x - 4x + 1) = 0 The GCF is x. x(x - 1)(x - 1) = 0 x = 0 or x - 1 = 0 or x - 1 = 0 x = 0 or x = 1 The solutions are 0 and 1. or x = 1 Subtract 4x from each side. Factor the perfect square trinomial. Use the Zero Product Property. Solve each equation. Online Connection Notice in Example 1b that the solution, or root, of 1 called a double root. It has multiplicity. appears twice. This is Lesson