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**Welcome to Interactive Chalkboard**

Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio Welcome to Interactive Chalkboard

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Splash Screen

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**Lesson 7-1 Polynomial Functions **

Lesson 7-2 Graphing Polynomial Functions Lesson 7-3 Solving Equations Using Quadratic Techniques Lesson 7-4 The Remainder and Factor Theorems Lesson 7-5 Roots and Zeros Lesson 7-6 Rational Zero Theorem Lesson 7-7 Operations on Functions Lesson 7-8 Inverse Functions and Relations Lesson 7-9 Square Root Functions and Inequalities Contents

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**Example 1 Find Degrees and Leading Coefficients **

Example 2 Evaluate a Polynomial Function Example 3 Functional Values of Variables Example 4 Graphs of Polynomial Functions Lesson 1 Contents

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**State the degree and leading coefficient of. in one variable**

State the degree and leading coefficient of in one variable. If it is not a polynomial in one variable, explain why. Answer: This is a polynomial in one variable. The degree is 3 and the leading coefficient is 7. Example 1-1a

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**State the degree and leading coefficient of. in one variable**

State the degree and leading coefficient of in one variable. If it is not a polynomial in one variable, explain why. Answer: This is not a polynomial in one variable. It contains two variables, a and b. Example 1-1b

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**State the degree and leading coefficient of. in one variable**

State the degree and leading coefficient of in one variable. If it is not a polynomial in one variable, explain why. Answer: This is not a polynomial in one variable. The term 2c–1 is not of the form ancn, where n is a nonnegative integer. Example 1-1c

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**Rewrite the expression so the powers of y are in decreasing order.**

State the degree and leading coefficient of in one variable. If it is not a polynomial in one variable, explain why. Rewrite the expression so the powers of y are in decreasing order. Answer: This is a polynomial in one variable with degree of 4 and leading coefficient 1. Example 1-1d

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**Answer: degree 3, leading coefficient 3**

State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. a. b. Answer: degree 3, leading coefficient 3 Answer: This is not a polynomial in one variable. It contains two variables, x and y. Example 1-1e

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**Answer: degree 3, leading coefficient 1**

Answer: This is not a polynomial in one variable. The term 3a–1 is not of the form ancn, where n is nonnegative. Answer: degree 3, leading coefficient 1 Example 1-1f

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**Find the values of f (4), f (5), and f (6).**

Nature Refer to Example 2 on page 347 of your textbook. A sketch of the arrangement of hexagons shows a fourth ring of 18 hexagons, a fifth ring of 24 hexagons, and a sixth ring of 30 hexagons. Show that the polynomial function gives the total number of hexagons when Find the values of f (4), f (5), and f (6). Original function Replace r with 4. Simplify. Example 1-2a

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**Original function Replace r with 5. Simplify. Original function**

Example 1-2b

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**Answer: These match the function values for respectively.**

From the information given in Example 2 of your textbook, you know that the total number of hexagons for three rings is 19. So, the total number of hexagons for four rings is or 37, five rings is or 61, and six rings is or 91. Answer: These match the function values for respectively. Example 1-2c

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**Find the total number of hexagons in a honeycomb with 20 rings.**

Nature Refer to Example 2 on page 347 of your textbook. A sketch of the arrangement of hexagons shows a fourth ring of 18 hexagons, a fifth ring of 24 hexagons, and a sixth ring of 30 hexagons. Find the total number of hexagons in a honeycomb with 20 rings. Original function Replace r with 20. Answer: Simplify. Example 1-2d

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Nature A sketch of the arrangement of hexagons shows a seventh ring of 36 hexagons, an eighth ring of 42 hexagons, and a ninth ring of 48 hexagons. a. Show that the polynomial function gives the total number of hexagons when Recall that the total number of hexagons in six rings is 91. Answer: f (7) = 127; f (8) = 169; f (9) = 217; the total number of hexagons for seven rings is or 127, eight rings is or 169, and nine rings is or 217. These match the functional values for r = 7, 8, and 9, respectively. Example 1-2e

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**b. Find the total number of hexagons in a honeycomb with 30 rings.**

Answer: 2611 Example 1-2f

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**Find Original function Replace x with y 3. Answer: Property of powers**

Example 1-3a

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**To evaluate b(2x – 1), replace m in b(m) with 2x – 1.**

Find To evaluate b(2x – 1), replace m in b(m) with 2x – 1. Original function Replace m with 2x – 1. Evaluate 2(2x – 1)2. Simplify. Example 1-3b

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**Distributive Property**

To evaluate 3b(x), replace m with x in b(m), then multiply the expression by 3. Original function Replace m with x. Distributive Property Example 1-3c

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**Now evaluate b(2x – 1) – 3b(x).**

Replace b(2x – 1) and 3b(x) with evaluated expressions. Simplify. Answer: Example 1-3d

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a. Find b. Find Answer: Answer: Example 1-3e

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** describe the end behavior, **

For the graph, describe the end behavior, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros. Answer: It is an even-degree polynomial function. The graph does not intersect the x-axis, so the function has no real zeros. . Example 1-4a

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** describe the end behavior, **

For the graph, describe the end behavior, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros. Answer: It is an odd-degree polynomial function. The graph intersects the x-axis at one point, so the function has one real zero. . Example 1-4b

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** describe the end behavior, **

For the graph, describe the end behavior, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros. Answer: It is an even-degree polynomial function. The graph intersects the x-axis at two points, so the function has two real zeros. . Example 1-4c

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** describe the end behavior, **

For each graph, a. describe the end behavior, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros. Answer: It is an even-degree polynomial function. The graph intersects the x-axis at two points, so the function has two real zeros. . Example 1-4d

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** describe the end behavior, **

For each graph, b. describe the end behavior, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros. Answer: It is an odd-degree polynomial function. The graph intersects the x-axis at three points, so the function has three real zeros. . Example 1-4e

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** describe the end behavior, **

For each graph, c. describe the end behavior, determine whether it represents an odd-degree or an even-degree function, and state the number of real zeros. Answer: It is an even-degree polynomial function. The graph intersects the x-axis at one point, so the function has one real zero. . Example 1-4f

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End of Lesson 1

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**Example 1 Graph a Polynomial Function **

Example 2 Locate Zeros of a Function Example 3 Maximum and Minimum Points Example 4 Graph a Polynomial Model Lesson 2 Contents

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**making a table of values. –4 5 –3 –2 –1 2**

Graph by making a table of values. x f(x) –4 5 –3 –2 –1 2 1 –19 Answer: Example 2-1a

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**making a table of values.**

Graph by making a table of values. This is an odd degree polynomial with a negative leading coefficient, so f (x) + as x – and f (x) – as x +. Notice that the graph intersects the x-axis at 3 points indicating that there are 3 real zeros. Answer: Example 2-1b

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**making a table of values. –3 –8 –2 1 –1 2**

Graph by making a table of values. x f (x) –3 –8 –2 1 –1 2 4 17 Answer: Example 2-1c

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Determine consecutive values of x between which each real zero of the function is located. Then draw the graph. Make a table of values. Since f (x) is a 4th degree polynomial function, it will have between 0 and 4 zeros, inclusive. x f (x) –2 9 –1 1 –3 2 –7 3 19 change in signs change in signs change in signs change in signs Example 2-2a

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**Look at the value of f (x) to locate the zeros**

Look at the value of f (x) to locate the zeros. Then use the points to sketch the graph of the function. Answer: There are zeros between x = –2 and –1, x = –1 and 0, x = 0 and 1, and x = 2 and 3. Example 2-2b

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**There are zeros between x = –1 and 0, x = 0 and 1, and x = 3 and 4.**

Determine consecutive values of x between which each real zero of the function is located. Then draw the graph. Answer: There are zeros between x = –1 and 0, x = 0 and 1, and x = 3 and 4. Example 2-2c

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**Make a table of values and graph the function.**

Graph Estimate the x-coordinates at which the relative maximum and relative minimum occur. Make a table of values and graph the function. x f (x) –2 –19 –1 5 1 2 –3 3 –4 4 30 zero at x = –1 indicates a relative maximum zero between x = 1 and x = 2 indicates a relative minimum zero between x = 3 and x = 4 Example 2-3a

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Answer: The value of f (x) at x = 0 is greater than the surrounding points, so it is a relative maximum. The value of f (x) at x = 3 is less than the surrounding points, so it is a relative minimum. x f (x) –2 –19 –1 5 1 2 –3 3 –4 4 30 Example 2-3b

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**Graph Estimate the x-coordinates at which the relative maximum and relative minimum occur.**

Answer: The value of f (x) at x = 0 is less than the surrounding points, so it is a relative minimum. The value of f (x) at x = –2 is greater than the surrounding points, so it is a relative maximum. Example 2-3c

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Health The weight w, in pounds, of a patient during a 7-week illness is modeled by the cubic equation where n is the number of weeks since the patient became ill. Graph the equation. Make a table of values for weeks 0 through 7. Plot the points and connect with a smooth curve. Example 2-4a

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n w(n) 110 1 109.5 2 108.4 3 107.3 4 106.8 5 107.5 6 7 114.9 Answer: Example 2-4b

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**Describe the turning points of the graph and its end behavior.**

Answer: There is a relative minimum at week 4. For the end behavior, w (n) increases as n increases. Example 2-4c

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**What trends in the patient’s weight does the graph suggest?**

Answer: The patient lost weight for each of 4 weeks after becoming ill. After 4 weeks, the patient started to gain weight and continues to gain weight. Example 2-4d

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Weather The rainfall r, in inches per month, in a Midwestern town during a 7-month period is modeled by the cubic equation where m is the number of months after March 1. a. Graph the equation. Answer: Example 2-4e

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**b. Describe the turning. points of the graph. and its end behavior. c**

b. Describe the turning points of the graph and its end behavior c. What trends in the amount of rainfall received by the town does the graph suggest? Answer: There is a relative maximum at Month 2, or May. For the end behavior, r (m) decreases as m increases. Answer: The rainfall increased for two months following March. After two months, the amount of rainfall decreased for the next five months and continues to decrease. Example 2-4f

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End of Lesson 2

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**Example 1 Write an Expression in Quadratic Form **

Example 2 Solve Polynomial Equations Example 3 Solve Equations with Rational Exponents Example 4 Solve Radical Equations Lesson 3 Contents

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**Write in quadratic form, if possible.**

Answer: Example 3-1a

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**Write in quadratic form, if possible.**

Answer: Example 3-1b

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**Write in quadratic form, if possible.**

Answer: This cannot be written in quadratic form since Example 3-1c

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**Write in quadratic form, if possible.**

Answer: Example 3-1d

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**Write each expression in quadratic form, if possible. a. b. c.**

Answer: Answer: Answer: This cannot be written in quadratic form since Answer: Example 3-1e

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**Write the expression on the left in quadratic form.**

Solve . Original equation Write the expression on the left in quadratic form. Factor the trinomial. Factor each difference of squares. Example 3-2a

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**Use the Zero Product Property.**

or Answer: The solutions are –5, –2, 2, and 5. Example 3-2b

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**Check. The graph of. shows**

Check The graph of shows that the graph intersects the x-axis at –5, –2, 2, and 5. Example 3-2c

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**This is the sum of two cubes.**

Solve . Original equation This is the sum of two cubes. Sum of two cubes formula with a = x and b = 6 or Zero Product Property Example 3-2d

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**Replace a with 1, b with –6, and c with 36.**

The solution of the first equation is –6. The second equation can be solved by using the Quadratic Formula. Quadratic Formula Replace a with 1, b with –6, and c with 36. Simplify. Example 3-2e

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**Answer: The solutions of the original equation are**

Simplify. Answer: The solutions of the original equation are Example 3-2f

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Solve each equation. a. b. Answer: –3, –1, 1, 3 Answer: Example 3-2g

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**Write the expression on the left in quadratic form.**

Solve Original equation Write the expression on the left in quadratic form. Factor the trinomial. Zero Product Property or Example 3-3a

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**Isolate x on one side of the equation. or**

Raise each side to the fourth power. Simplify. Example 3-3b

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**Check Substitute each value into the original equation.**

Answer: The solution is 81. Example 3-3c

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Solve Answer: –8, –27 Example 3-3d

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**Rewrite so that one side is zero.**

Solve Original equation Rewrite so that one side is zero. Write the expression on the left side in quadratic form. Factor. Example 3-4a

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**Zero Product Property or Solve each equation.**

Answer: Since the square root of x cannot be negative, the equation has no solution. Thus, the only solution of the original equation is 9. Example 3-4b

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Solve Answer: 25 Example 3-4c

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End of Lesson 3

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**Example 1 Synthetic Substitution Example 2 Use the Factor Theorem **

Example 3 Find All Factors of a Polynomial Lesson 4 Contents

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**3 10 41 164 654 If find f (4). Method 1 Synthetic Substitution**

By the Remainder Theorem, f (4) should be the remainder when you divide the polynomial by x – 4. Notice that there is no x term. A zero is placed in this position as a placeholder. Answer: The remainder is 654. Thus, by using synthetic substitution, f (4) = 654. Example 4-1a

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**Method 2 Direct Substitution**

Replace x with 4. Original function Replace x with 4. Simplify. or 654 Answer: By using direct substitution, f (4) = 654. Example 4-1b

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If find f (3). Answer: 34 Example 4-1c

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**Show that is a factor of Then find the remaining factors of the polynomial.**

The binomial x – 3 is a factor of the polynomial if 3 is a zero of the related polynomial function. Use the factor theorem and synthetic division. Example 4-2a

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**Since the remainder is 0, (x – 3) is a factor of the polynomial**

Since the remainder is 0, (x – 3) is a factor of the polynomial. The polynomial can be factored as The polynomial is the depressed polynomial. Check to see if this polynomial can be factored. Factor the trinomial. Answer: So, Example 4-2b

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**Check. You can see that the graph of the related function**

Check You can see that the graph of the related function crosses the x-axis at 3, –6, and –1. Thus, Example 4-2c

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**Show that is a factor of Then find the remaining factors of the polynomial.**

Answer: So, Since Example 4-2d

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**Geometry The volume of a rectangular prism is given by Find the missing measures.**

The volume of a rectangular prism is You know that one measure is x – 2, so x – 2 is a factor of V(x). Example 4-3a

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**The quotient is . Use this to factor V(x).**

Volume function Factor. Factor the trinomial Answer: The missing measures of the prism are x + 4 and x + 5. Example 4-3b

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**Answer: The missing measures of the prism are x – 2 and x + 5.**

Geometry The volume of a rectangular prism is given by Find the missing measures. Answer: The missing measures of the prism are x – 2 and x + 5. Example 4-3c

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End of Lesson 4

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**Example 1 Determine Number and Type of Roots **

Example 2 Find Numbers of Positive and Negative Zeros Example 3 Use Synthetic Substitution to Find Zeros Example 4 Use Zeros to Write a Polynomial Function Lesson 5 Contents

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**Solve State the number and type of roots.**

Original equation Add 10 to each side. Answer: This equation has exactly one real root, 10. Example 5-1a

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**Solve State the number and type of roots.**

Original equation Factor. Zero Product Property or Solve each equation. Answer: This equation has two real roots, –8 and 6. Example 5-1b

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**Solve State the number and type of roots.**

Original equation Factor out the GCF. Use the Zero Product Property. or Subtract 6 from each side. Example 5-1c

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Square Root Property Answer: This equation has one real root at 0, and two imaginary roots at Example 5-1d

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**Solve State the number and type of roots.**

Original equation Factor differences of squares. Factor differences of squares. or Zero Product Property Example 5-1e

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Solve each equation. Answer: This equation has two real roots, –2 and 2, and two imaginary roots, 2i and –2i. Example 5-1f

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**Solve each equation. State the number and type of roots.**

Answer: This equation has exactly one root at –3. Answer: This equation has exactly two roots, –3 and 4. Answer: This equation has one real root at 0 and two imaginary roots at Example 5-1g

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d. Answer: This equation has two real roots, –3 and 3, and two imaginary roots, 3i and –3i. Example 5-1h

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**State the possible number of positive real zeros, negative real zeros, and imaginary zeros of**

Since p(x) has degree 6, it has 6 zeros. However, some of them may be imaginary. Use Descartes Rule of Signs to determine the number and type of real zeros. Count the number of changes in sign for the coefficients of p(x). yes – to + yes + to – no – to – no – to – Example 5-2a

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**Since there are two sign changes, there are 2 or 0 positive real zeros**

Since there are two sign changes, there are 2 or 0 positive real zeros. Find p(–x) and count the number of sign changes for its coefficients. x 1 no – to – no – to – yes – to + yes + to – Since there are two sign changes, there are 2 or 0 negative real zeros. Make a chart of possible combinations. Example 5-2b

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**Answer: 2 6 4 Number of Positive Real Zeros**

Number of Negative Real Zeros Number of Imaginary Zeros Total 2 6 4 Example 5-2c

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**State the possible number of positive real zeros, negative real zeros, and imaginary zeros of**

Answer: The function has either 2 or 0 positive real zeros, 2 or 0 negative real zeros, and 4, 2, or 0 imaginary zeros. Example 5-2d

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Find all of the zeros of Since f (x) has degree of 3, the function has three zeros. To determine the possible number and type of real zeros, examine the number of sign changes in f (x) and f (–x). yes yes no no no yes Example 5-3a

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The function has 2 or 0 positive real zeros and exactly 1 negative real zero. Thus, this function has either 2 positive real zeros and 1 negative real zero or 2 imaginary zeros and 1 negative real zero. To find the zeros, list some possibilities and eliminate those that are not zeros. Use a shortened form of synthetic substitution to find f (a) for several values of a. x 1 –1 2 4 –3 –4 14 –38 –2 8 –12 Each row in the table shows the coefficients of the depressed polynomial and the remainder. Example 5-3b

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**Replace a with 1, b with –2, and c with 4.**

From the table, we can see that one zero occurs at x = –1. Since the depressed polynomial, , is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation Quadratic Formula Replace a with 1, b with –2, and c with 4. Example 5-3c

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Simplify. Simplify. Example 5-3d

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Answer: Thus, this function has one real zero at –1 and two imaginary zeros at and The graph of the function verifies that there is only one real zero. Example 5-3e

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Find all of the zeros of Answer: Example 5-3f

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**Short-Response Test Item **

Write a polynomial function of least degree with integer coefficients whose zeros include 4 and 4 – i. Read the Test Item • If 4 – i is a zero, then 4 + i is also a zero, according to the Complex Conjugate Theorem. So, x – 4, x – (4 – i), and x – (4 + i) are factors of the polynomial function. Example 5-4a

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**• Multiply the factors to find the polynomial function. **

Solve the Test Item • Write the polynomial function as a product of its factors. • Multiply the factors to find the polynomial function. Write an equation. Regroup terms. Rewrite as the difference of two squares. Example 5-4b

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**Square x – 4 and replace i2 with –1.**

Simplify. Multiply using the Distributive Property. Combine like terms. Example 5-4c

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Answer: is a polynomial function of least degree with integral coefficients whose zeros are 4, 4 – i, and 4 + i. Example 5-4d

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**Short-Response Test Item **

Write a polynomial function of least degree with integer coefficients whose zeros include 2 and 1 + i. Answer: Example 5-4e

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End of Lesson 5

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**Example 1 Identify Possible Zeros **

Example 2 Use the Rational Zero Theorem Example 3 Find All Zeros Lesson 6 Contents

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**List all of the possible rational zeros of**

If is a rational zero, then p is a factor of 4 and q is a factor of 3. The possible factors of p are 1, 2, and 4. The possible factors of q are 1 and 3. Answer: So, Example 6-1a

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**List all of the possible rational zeros of**

Since the coefficient of x4 is 1, the possible zeros must be a factor of the constant term –15. Answer: So, the possible rational zeros are 1, 3, 5, and 15. Example 6-1b

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**List all of the possible rational zeros of each function. a.**

Answer: Answer: Example 6-1c

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**Let x = the height, x – 2 = the width, and x + 4 = the length.**

Geometry The volume of a rectangular solid is 1120 cubic feet. The width is 2 feet less than the height and the length is 4 feet more than the height. Find the dimensions of the solid. Let x = the height, x – 2 = the width, and x + 4 = the length. Example 6-2a

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**Write the equation for volume.**

Formula for volume Multiply. Subtract 1120. The leading coefficient is 1, so the possible integer zeros are factors of Since length can only be positive, we only need to check positive zeros. Example 6-2b

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The possible factors are 1, 2, 4, 5, 8, 10, 14, 16, 20, 28, 32, 35, 40, 56, 70, 80, 112, 140, 160, 224, 280, 560, and By Descartes’ Rule of Signs, we know that there is exactly one positive real root. Make a table and test possible real zeros. p 1 2 –8 –1120 3 –5 –1125 6 –1112 10 12 112 So, the zero is 10. The other dimensions are 10 – 2 or 8 feet and or 14 feet. Example 6-2c

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**Check Verify that the dimensions are correct.**

Answer: Example 6-2d

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**Geometry The volume of a rectangular solid is 100 cubic feet**

Geometry The volume of a rectangular solid is 100 cubic feet. The width is 3 feet less than the height and the length is 5 feet more than the height. Find the dimensions of the solid. Answer: Example 6-2e

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**The possible rational zeros are 1, 2, 3, 5, 6, 10, 15, and 30. **

Find all of the zeros of From the corollary to the Fundamental Theorem of Algebra, we know there are exactly 4 complex roots. According to Descartes’ Rule of Signs, there are 2 or 0 positive real roots and 2 or 0 negative real roots. The possible rational zeros are 1, 2, 3, 5, 6, 10, 15, and 30. Make a table and test some possible rational zeros. Example 6-3a

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–15 –13 3 1 2 24 –6 –17 30 11 –19 p q Since f (2) = 0, you know that x = 2 is a zero. The depressed polynomial is Example 6-3b

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**There is another zero at x = 3. The depressed polynomial is**

Since x = 2 is a positive real zero, and there can only be 2 or 0 positive real zeros, there must be one more positive real zero. Test the next possible rational zeros on the depressed polynomial. 5 6 1 3 –15 –13 p q There is another zero at x = 3. The depressed polynomial is Example 6-3c

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**Write the depressed polynomial.**

Factor Write the depressed polynomial. Factor. or Zero Product Property There are two more real roots at x = –5 and x = –1. Answer: The zeros of this function are –5, –1, 2, and 3. Example 6-3d

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Find all of the zeros of Answer: –5, –3, 1, and 3 Example 6-3e

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End of Lesson 6

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**Example 1 Add and Subtract Functions **

Example 2 Multiply and Divide Functions Example 3 Evaluate Composition of Relations Example 4 Simplify Composition of Functions Example 5 Use Composition of Functions Lesson 7 Contents

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Given , find Addition of functions and Simplify. Answer: Example 7-1a

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**Subtraction of functions**

Given , find Subtraction of functions and Simplify. Answer: Example 7-1b

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**Given find each function.**

b. Answer: Answer: Example 7-1c

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**Distributive Property**

Given find Product of functions and Distributive Property Distributive Property Simplify. Answer: Example 7-2a

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Given find Division of functions Answer: and Example 7-2b

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**Since 4 makes the denominator 0, it is excluded from the domain of**

Example 7-2c

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**Given find each function.**

b. Answer: Answer: Example 7-2d

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**If f (x) = {(2, 6), (9, 4), (7, 7), (0, –1)} and g (x) = {(7, 0), (–1, 7), (4, 9), (8, 2)}, find and**

To find , evaluate g (x) first. Then use the range of g as the domain of f and evaluate f (x). Answer: Example 7-3a

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**Answer: Since 6 is not in the domain of g, is undefined for x = 2.**

To find evaluate f (x) first. Then use the range of f as the domain of g and evaluate g (x). is undefined. Answer: Since 6 is not in the domain of g, is undefined for x = 2. Example 7-3b

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**If f (x) = {(1, 2), (0, –3), (6, 5), (2, 1)} and g (x) = {(2, 0), (–3, 6), (1, 0), (6, 7)}, find and**

Answer: Example 7-3c

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**Composition of functions**

Find and and Composition of functions Replace g (x) with 2x – 1. Substitute 2x – 1 for x in f (x). Example 7-4a

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**Composition of functions**

Evaluate (2x – 1)2. Simplify. Composition of functions Replace f (x) with Example 7-4b

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**Substitute for x in g (x).**

Simplify. Answer: So, and Example 7-4c

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**Evaluate and x = –2. Function from part a Replace x with –2. Simplify.**

Example 7-4d

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**Function from part a Replace x with –2. Simplify. Answer: So, and**

Example 7-4e

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**a. Find and and b. Evaluate and x = 1. Answer: and Answer: and**

Example 7-4f

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**Taxes Tracie Long has $100 deducted from every paycheck for retirement**

Taxes Tracie Long has $100 deducted from every paycheck for retirement. She can have this deduction taken before state taxes are applied, which reduces her taxable income. Her state income tax is 4%. If Tracie earns $1500 every pay period, find the difference in her net income if she has the retirement deduction taken before or after state taxes. Explore Let x = her income per paycheck, r (x) = her income after the deduction for retirement, t (x) = her income after tax. Example 7-5a

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**$100 is deducted for retirement.**

Plan Write equations for r (x) and t (x). $100 is deducted for retirement. The tax rate is 4%. Solve If Tracie has her retirement deducted before taxes, then her net income is represented by Replace x with in Example 7-5b

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Replace x with in If Tracie has her retirement deducted after taxes, then her net income is represented by Replace x with 1500 in Example 7-5c

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Replace x with in Answer: and The difference is 1344 – 1340 or 4. So, her net income is $4 more if the retirement deduction is taken before taxes. Examine The answer makes sense. Since the taxes are being applied to a smaller amount, less taxes will be deducted from her paycheck. Example 7-5d

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Taxes Brandi Smith has $200 deducted from every paycheck for retirement. She can have this deduction taken before state taxes are applied, which reduces her taxable income. Her state income tax is 10%. If Brandi earns $2200 every pay period, find the difference in her net income if she has the retirement deduction taken before or after state taxes. Answer: Her net income is $20 more if she has the retirement deduction taken before her state taxes. Example 7-5e

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End of Lesson 7

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**Example 1 Find an Inverse Relation Example 2 Find an Inverse Function **

Example 3 Verify Two Functions are Inverses Lesson 8 Contents

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Geometry The ordered pairs of the relation {(1, 3), (6, 3), (6, 0), (1, 0)} are the coordinates of the vertices of a rectangle. Find the inverse of this relation and determine whether the resulting ordered pairs are also the coordinates of the vertices of a rectangle. To find the inverse of this relation, reverse the coordinates of the ordered pairs. The inverse of the relation is {(3, 1), (3, 6), (0, 6), (0, 1)}. Example 8-1a

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Answer: Plotting the points shows that the ordered pairs also describe the vertices of a rectangle. Notice that the graph of the relation and the inverse are reflections over the graph of y = x. Example 8-1b

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Geometry The ordered pairs of the relation {(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)} are the coordinates of the vertices of a pentagon. Find the inverse of this relation and determine whether the resulting ordered pairs are also the coordinates of the vertices of a pentagon. Answer: {(4, –3), (5, –1), (3, 2), (1, 1), (1, –2)} These ordered pairs also describe the vertices of a pentagon. Example 8-1c

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**Step 1 Replace f (x) with y in the original equation.**

Find the inverse of Step 1 Replace f (x) with y in the original equation. Step 2 Interchange x and y. Example 8-2a

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**Step 4 Replace y with f –1(x).**

Step 3 Solve for y. Inverse Multiply each side by –2. Add 2 to each side. Step 4 Replace y with f –1(x). Example 8-2b

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**Answer: The inverse of is**

Example 8-2c

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**Graph the function and its inverse.**

Graph both functions on the coordinate plane. The graph of is the reflection for over the line Example 8-2d

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Answer: Example 8-2e

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**b. Graph the function and its inverse. Answer:**

a. Find the inverse of b. Graph the function and its inverse. Answer: Answer: Example 8-2f

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**Determine whether and are inverse functions.**

Check to see if the compositions of f (x) and g (x) are identity functions. Example 8-3a

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**Answer: The functions are inverses since both and equal x.**

Example 8-3b

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**Determine whether and are inverse functions.**

Answer: The functions are inverses since both compositions equal x. Example 8-3c

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End of Lesson 8

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**Example 1 Graph a Square Root Function **

Example 2 Solve a Square Root Problem Example 3 Graph a Square Root Inequality Lesson 9 Contents

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**Graph State the domain, range, and x- and y-intercepts.**

Since the radicand cannot be negative, identify the domain. Write the expression inside the radicand as 0. Solve for x. The x-intercept is Example 9-1a

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**Make a table of values to graph the function.**

2.83 6 2.55 5 2.23 4 1.87 3 1.14 2 0.71 1 y x Example 9-1b

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Answer: From the graph, you can see that the domain is and the range is y 0. The x-intercept is There is no y-intercept. Example 9-1c

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**x-intercept: 1 y-intercept: none**

Graph State the domain, range, and x- and y-intercepts. Answer: domain: x 1 range: y 0 x-intercept: 1 y-intercept: none Example 9-1d

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**Graph the function. State the domain and range.**

Physics When an object is spinning in a circular path of radius 2 meters with velocity v, in meters per second, the centripetal acceleration a, in meters per second squared, is directed toward the center of the circle. The velocity v and acceleration a of the object are related by the function Graph the function. State the domain and range. Example 9-2a

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**The function is Make a table of values and graph the function.**

Answer: The domain is a 0 and the range is v 0. a v 1 1.41 2 3 2.45 4 2.83 5 3.16 Example 9-2b

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What would be the centripetal acceleration of an object spinning along the circular path with a velocity of 4 meters per second? Original equation Replace v with 4. Square each side. Divide each side by 2. Answer: The centripetal acceleration would be 8 meters per second squared. Example 9-2c

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**a. Graph the function. State the domain and range.**

Geometry The volume V and surface area A of a soap bubble are related by the function a. Graph the function. State the domain and range. Answer: domain: A 0 range: V 0 Example 9-2d

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**b. What would the surface area be if the volume were 94 cubic units?**

Answer: 100 units2 Example 9-2e

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Graph Graph the related equation Since the boundary is not included, the graph should be dashed. Example 9-3a

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**The domain includes values for So the graph is to the right of**

Select a point and test its ordered pair. Test (0, 0). false Shade the region that does not include (0, 0). Example 9-3b

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**Graph the related equation**

The domain includes values for Select a point and test its ordered pair. Example 9-3c

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**Shade the region that includes (4, 1).**

Test (4, 1). true Shade the region that includes (4, 1). Example 9-3d

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a. Graph Answer: Example 9-3e

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b. Graph Answer: Example 9-3f

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End of Lesson 9

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**Explore online information about the information introduced in this chapter.**

Click on the Connect button to launch your browser and go to the Algebra 2 Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to Algebra2.com

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4.7 Complete the Square.

4.7 Complete the Square.

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