ready to go on? skills intervention 6a 6-1 polynomialsready to go on? skills intervention 6-1...

Copyright © by Holt, Rinehart and Winston. 91 Holt Algebra 2All rights reserved.

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SECTION

6AReady To Go On? Skills Intervention6-1 Polynomials

Find these vocabulary words in Lesson 6-1 and the Multilingual Glossary.

Classifying PolynomialsRewrite the polynomial 4 � 7x 3 � 2 x 2 � 9x in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial.

A polynomial is written in standard form when its terms are written in order by degree.

leading coefficient

7 x 3 � x 2 � x � Write the given polynomial in standard form.

degree of polynomial

Identify the leading coefficient. Identify the degree of the leading coefficient.

A term is separated by addition or subtraction signs. How many terms are there?

A polynomial is named by its largest degree. Is the polynomial quadratic or cubic?

Adding and Subtracting PolynomialsAdd or subtract. Write your answer in standard form.

A. (9 x 2 � x 3 � 4) � (3 x 3 � 6 x 2 � 3x)

To add vertically, write each polynomial in form and align terms.

x 3 � x 2 � 4 First polynomial

� x 3 � 6 x 2 � x Second polynomial

x 3 � x 2 � 3x � Add.

B. (10 x 3 � 5 x 2 ) � (3 x 3 � x 2 � 5)

To subtract horizontally, you add the .

(10 x 3 � 5 x 2 ) � (� x 3 � x 2 � 5) Change the sign and distribute the negative.

(10 x 3 � x 3 ) � ( x 2 � x 2 ) � Rearrange so like terms are together.

x 3 � x 2 � 5 Add.

Vocabulary

monomial polynomial degree of a monomial degree of a polynomial

leading coefficient binomial trinomial polynomial function

Multiplying PolynomialsFind each product.

To multiply any two polynomials, use the Property to multiply each

term in the second polynomial by each term in the polynomial.

A. Multiply horizontally. (2x � 1)( x 3 � 2 x 2 � 6x � 1)

Distribute 2x, and then �1.

2x( x 3 ) � (2 x 2 ) � (6x) � 2x(1) � 1( x 3 ) � (2 x 2 ) � 1 x � 1(1)

x 4 � 4 x 3 � x 2 � 2x � � 2 x 2 � x � 1 When you multiply, you

exponents.

x 4 � ( x 3 � x 3 ) � (12 x 2 � x 2 ) � ( x � 6x) � 1 Combine like terms.

x 4 � 5 x 3 � x 2 � x � 1 Simplify.

B. Multiply vertically. ( x 2 � 6x � 5)(x � 4)

x 2 � 6x � 5

x � 4

� x 2 � x � 20 Multiply x 2 � 6x � 5 by �4.

x 3 � x 2 � x Multiply x 2 � 6x � 5 by x.

x 3 � x 2 � x � Combine like terms.

Using Pascal’s Triangle to Expand Binomial ExpressionsExpand the expression (3x � 4 ) 3 .

Each row of Pascal’s triangle gives the of the corresponding binomial expansion.

Using Pascal’s Triangle, what are the coefficients for n � 3 (row 4)?

The exponents on the first term and the exponents on the second term decrease.What is the first term in the expression (3x � 4 ) 3 ? second term?

Complete the expansion.

[ 1(3x ) 3 (�4 ) 0 ] � [ 3(3x ) 2 (�4 ) ] � [ (3x ) 1 (�4 ) ] � [ 1(3x ) (�4 ) ] [ x 3 (1) ] � [ 3 x 2 (�4) ] � [ x( ) ] � [ 1(1) ] Simplify.

x 3 � x 2 � x � 64 Simplify.

Pascal’s Triangle1

1 1

1 2 1

1 3 3 1

1 4 6 4 1


Name Date Class

Ready To Go On? Skills Intervention6-2 Multiplying Polynomials6A

SECTION

6AReady To Go On? Skills Intervention6-3 Dividing Polynomials


Name Date Class

SECTION

Find this vocabulary word in Lesson 6-3 and the Multilingual Glossary.

Using Synthetic Division to Divide by a Linear BinomialDivide. ( x 3 � 6 x 2 � x � 30) � (x � 2)

What are the coefficients of the dividend? 1, , ,

The divisor is written in the form x � a. What is the value of a ?

Write the coefficients of the dividend and the value for a in the upper left corner.

2 1 6 �1

1

2 1 6 �1

2 �

1

2 1 6 �1

2 16

1 0

Bring down the first coefficient (1) and write it below the horizontal bar.

Multiply 2 by 1 to get 2. Write the product under the next coefficient and add.

Repeat the steps (multiply, write the product under the next coefficient and add) with the remaining numbers.

The answer is x 2 � � .

Using Synthetic SubstitutionUse synthetic substitution to evaluate P (x) � x 4 � 2 x 2 � 2x � 5 for x � 2.

You can use synthetic division to evaluate polynomials. The process is exactly the same as synthetic division, but the final answer is interpreted differently.

If you are missing terms what do you do?

What are the coefficients of the polynomial? 1, , 2, ,

Write the coefficients.

2 1 2 �5Bring down the first coefficient (1) and write it below the horizontal bar.

2 12

6Complete the synthetic division by multiplying and adding.

Check: Substitute 2 for x in P(x) � x 4 � 2 x 2 � 2x � 5. The answer should be your remainder.

P(x ) � 2 4 � 2 � � 2 � 2 � � � 5

� 16 � � � 5

� Does your answer check?

Vocabulary

synthetic division

�

Factoring by GroupingFactor x 3 � 6x 2 � 3x � 18.

� x 3 � x 2 � � � x � 18 � Group the terms.

The common monomial for the first group is x 2, what is the common

monomial for the second group?

x 2 � x � � � � x � � Factor common monomials from each group.

What is the common binomial factor?

� x � � � x 2 � � Factor out the common binomial.

Factoring the Sum or Difference of Two CubesFactor each expression.

What is the general form for the sum of two cubes? a3 � b 3 � (a � )(a 2 � � b 2)

What is the general form for the difference of two cubes? a 3 � b 3 � (a � )(a 2 � � b 2)

A. 125 x 3 � y 3

Do you have a sum or difference of two cubes?

How can you write 125x 3 as a cube? � x � 3 How can you write y 3 as a cube? 3

Rewrite as a sum of cubes. � � 3 � 3

What represents a ? What represents b ?

Use the rule. a 3 � b 3 � � � y � � � � 2 � � � � � � y 2 � Simplify. � � � y � � � � y 2 � B. y 6 � 729

Do you have a sum or difference of two cubes?

How can you write y 6 as a cube? � y � 3 How can you write 729 as a cube? 3

What represents a ? What represents b ?

Rewrite as a difference of cubes. � � 3 � 3

Use the rule. a 3 � b 3 � � � 9 � � � y 2 � 2 � � � � � � 2 � Simplify. � � � 9 � � y 4 � � 81 � Factor y 2 � 9 as a difference of two squares.

� y � � � y � � � y 4 + � 81 �

6AReady To Go On? Skills Intervention6-4 Factoring Polynomials

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SECTION


Name Date Class

Ready To Go On? Quiz6A

SECTION

6-1 PolynomialsRewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial.

1. 7 x 2 � 4 x 5 � 3 2. 9 � 11x

Leading coefficient Leading coefficient

Degree Number of terms Degree Number of terms

Name Name

3. 2 � 6 x 3 � 2 x 2 � x 4. 7x � 3 x 4 � 6 x 3

Leading coefficient Leading coefficient

Degree Number of terms Degree Number of terms

Name Name

Add or subtract. Write your answer in standard form.

5. (4 x 2 � 3) � (5 x 2 � 4) 6. (10 x 3 � 7 x 2 ) � (3 x 3 � 2 x 2 � 4)

7. (12 x 2 � 2 x 3 + 9) � (6 x 3 � 5 x 2 � 3x)

8. The profit on x-units of a product can be modeled by P(x) � x 3 � 27x � 1500. Evaluate P (x) for x � 150, and describe what the value represents.

Graph each polynomial function on a calculator. Describe the graph, and identify the number of real zeros.

9. g (x ) � � 1 __ 2 x 5 � 2 x 2 10. h (x ) � 1 __ 4 x 3 � x 2 � 3

6-2 Multiplying PolynomialsFind each product.

11. 3y(2 x 2 � 5xy) 12. (a � 2b)(4ab � b 2 )

13. � 3x � 1 __ 4 � 2 14. (3x � 2)(2 x 3 � x 2 � 4x � 4)

Expand each expression.

15. (x � 4 ) 4 16. (x � 3y ) 3

17. (5x � 2 ) 4

18. Find the polynomial expression in terms of x for the volume of the rectangular prism shown.

6-3 Dividing PolynomialsDivide.

19. (10 x 2 � 13x � 3) � (5x � 1) 20. (4 x 3 � 13 x 2 � 12x � 4) � (x � 2)

Use synthetic substitution to evaluate the polynomial for the given value.

21. P(x) � 3 x 3 � 8 x 2 � 8x � 3 for x � 2 22. P(x) � x 5 � 5 x 4 � x � 6 for x � �5

6-4 Factoring Polynomials Factor each expression.

23. 4 y 3 � 24 y 2 � 28y 24. 9 y 2 � 25

25. 5 y 3 � 15 y 2 � 2y � 6 26. x 6 � 1

27. The volume of a box is modeled by the function V(x) � x 3 � 7 x 2 � 7x � 15. Identify the values of x for which the volume is 0 and use the graph to factor V(x).

x – 3 3x + 1

4x

66

121222

–151515155

–1–1212

–6–6– 22–22––66 44

y

x

55

–1–

–

6AReady To Go On? Quiz continued


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Ready To Go On? Enrichment6A

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SECTION

PackagingA package for a new toy is being designed out of a piece of heavy cardboard that is 421 square inches. The base has a width of 6 inches and the length is 6 inches more than the height. The piece of cardboard also has tabs that are 1 inch wide on each side of the ends and a 1-inch wide tab on the lid.

x

Top

Bottom

1

1

1

1

1

Find the height, x, and the volume of the box by following the steps.

1. Label all remaining dimensions in the figure.

2. Find an expression for each of the rectangular regions.

3. Write an expression for the surface area by adding together each of the rectangular areas.

SA �

�

4. Simplify the expression for surface area.

5. Write an equation for the surface area.

6. Solve the equation for x.

7. Find the approximate volume of the package.

xx

ToTopp

BottomBottom

11

11

11

11

11

Find this vocabulary word in Lesson 6-5 and the Multilingual Glossary.

Using Factoring to Solve Polynomial EquationsSolve the polynomial equation, 4 x 5 � 16 x 4 � 16 x 3 , by factoring.

4 x 5 � 16 x 4 � 16 x 3 What is the greatest common factor?

x 3 � x 2 � x � � Factor out the GCF.

x 3 � x � � � x � � Factor the quadratic.

� 0, x � 2 � , or x � � Set each factor equal to 0.

x � 0 x � x � Solve for x.

The roots are and .

Identifying MultiplicityIdentify the roots of each equation. State the multiplicity of each root.The multiplicity of a root is the number of times that x � r is a factor of P(x).

When a real root has an multiplicity, the graph touches the x-axis but does not cross it.

When a real root has an multiplicity greater than 1, the graph “bends” as it crosses the x-axis.

A. x 4 � 14 x 3 � 49 x 2 � 0

x 2 � 2

� 14x � � � 0 Factor out the greatest common factor.

� x � � � x � � � 0 Factor the quadratic.

x 2 � 0, x � � 0, or x � � 0 Set each factor equal to 0.

x � x � or x � Solve for x.

x is a factor times. The root 0 has multiplicity of .

x � 7 is a factor times. The root 7 has multiplicity of .

Check: Graph the function on a graphing calculator. Does the graph show that

y � x 4 � 14 x 3 � 49 x 2 touches or bends at the x-axis?

B. �2 x 3 � 10 x 2 � 16x � 96 � 0

� x 3 � x 2 � 8x � � � 0 Factor out the GCF.

(x � 3)(x � 4)(x � 4) Factor.

x � 3 is used as a factor time. The root �3 has a multiplicity of .

x � 4 is used as a factor times. The root has a multiplicity of .

Vocabulary

multiplicity

Ready To Go On? Skills Intervention6-5 Finding Real Roots of Polynomial Equations

Name Date Class

SECTION

6B


Name Date Class


Ready To Go On? Problem Solving Intervention6-5 Finding Real Roots of Polynomial Equations6B

SECTION

In a previous lesson you used several ways to factor polynomials. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots.

The yearly profit of a company in thousands of dollars can be modeled by P(t ) � t 4 � 74 t 2 � 1225, where t is the number of years since 2001. Factor the polynomial to find the years in which the profit was 0.

Understand the Problem

1. What needs to be done in order to find the year in which the profit is 0?

2. What is the given equation?

Make a Plan

3. How many roots will the equation have?

4. Write the equation when the profit is 0.

5. Is there a greatest common factor?

6. Is the polynomial a sum or difference of cubes?

7. Is the polynomial a difference of two squares?

8. What are the factors of 1225 that when added equal �74? and

Solve

9. Factor the polynomial.

( t 2 � )( t 2 � ) � 0

(t � )(t � )(t � )(t � ) � 0

t � 7 � 0, � 0, t � 5 � 0, � 0 Use the Zero Product Property.

t � , t � , t � , t � Solve for t.

10. What are the roots of the polynomial? , , ,

11. Can the negative roots be used? Why?

12. In what years will the profit be 0? 2001 � 5 � and 2001 � �

Look Back

13. Graph P(t ) � t 4 � 74 t 2 � 1225 on a graphing calculator.

14. Locate where the graph crosses the x-axis. Does it cross the x-axis at 5 and 7?

Writing Polynomial Functions Given ZerosWrite the simplest polynomial function with roots 3, �3, and 4.

If r is a root of P(x), then x � r is a factor of P(x).

P(x) � (x � 3)(x � )(x � ) Write each root as a factor.

� ( x 2 � � � 9)(x � ) Multiply the first two binomials.

� ( x 2 � )(x � ) Simplify.

� x 3 � x 2 � x � 36 Multiply the two binomials.

Finding All Roots of a Polynomial EquationSolve x 4 � 2 x 3 � x 2 � 8x � 12 � 0 by finding all roots.

The polynomial is of degree , so there are roots for the equation.Step 1 Use the Rational Root Theorem to identify all possible rational roots.

p � and q � 1 p __ q � �1, � , � , �4, �6, � __________________________

�1

Step 2 Graph y � x 4 � 2 x 3 � x 2 � 8x � 12 to find the roots.

The real roots appear to be at or near �1 and .Step 3 Test the possible real roots.

�1 1 �2 1 �8 �12

1 4 �12

3 1 –3 4 –12

12

1 0

The solutions are , , , and .

Writing a Polynomial Function with Complex RootsWrite the simplest polynomial function with zeros i, 2 and 0.

According to the Complex Conjugate Root Theorem, if a � bi is a root of a

polynomial equation with real-number coefficients, then is also a root.

There are four roots: , , 2 and 0.

P(x) � (x � i )(x � )(x � )x Write the equation in factored form.

� ( x 2 � � � i 2 )(x � )(x)

� ( x 2 � )( x 2 � x) Multiply and simplify.

P(x) � x 4 � x 3 � x 2 � x

The polynomial factors into (x � 1)( � 3 x 2 � � 12).Test the other root in the cubic polynomial.

The polynomial factors into (x � 1)(x � 3)( � ) � 0.

Step 4 Solve x 2 � � 0 to find the remaining roots.

x 2 � � 0

x 2 � � �


Name Date Class

Ready To Go On? Skills Intervention6-6 Fundamental Theorem of Algebra6B

SECTION

Name Date Class


Ready To Go On? Skills Intervention6-7 Investigating Graphs of Polynomial Functions6B

SECTION

Using Graphs to Analyze Polynomial FunctionsIdentify whether the function graphed has odd or even degree and a positive or negative lead coefficient.A polynomial with an odd degree and a leading coefficient a � 0:As x ��, P (x) �� and x ��, P (x) ��A polynomial with an even degree and a leading coefficient a � 0:As x ��, P (x) �� and x ��, P (x) ��A polynomial with an odd degree and a leading coefficient a � 0: As x ��, P (x) �� and x ��, P (x) ��A polynomial with an even degree and a leading coefficient a � 0:As x ��, P (x) �� and x ��, P (x) ��

Using the given graph: As x ��, P (x) and x ��, P (x)

P(x) is of degree with a leading coefficient.

Graphing a Polynomial FunctionGraph the function f(x) � x 3 � 5 x 2 � 2x � 8.

Step 1 When p � , q � �1, possible roots are: �1, , , and .

Step 2 Using synthetic division test possible rational zeros until a zero is identified.

Test x � �1. Test x � �2.

�1 1 5 2 �8

�4

1

�2 1 5 2

1 3

x � is a zero. So f (x) � (x � 2)( x 2 � x � ).

Step 3 Factor: (x � 2)(x � )(x � 1)

The zeros are �2, , and .

Step 4 To determine the y-intercept substitute for x.

f (0) � 0 3 � 5 � � 2 � 2 � � � 8 �

Step 5 Plot the zeros and y-intercept.Plot points between the zeros. Choose x � �3 and x � �1.

f (�3) � � 3 3 � 5 � � 2 � 2( ) � 8 � Plot the ordered pair (�3, ).

f (�1) � – 1 3 � 5 � � 2 � 2( ) � 8 � Plot the ordered pair (�1, ).

Step 6 Identify the end behavior:

As x ��, P(x) and x ��, P(x) .

Step 7 Sketch the graph of f (x) � x 3 � 5 x 2 � 2x � 8.

x

y

4

8

–8

–4–2–4–6 2 4


Name Date Class

Ready To Go On? Skills Intervention6-8 Transforming Polynomial Functions6B

SECTION

Reflecting Polynomial FunctionsLet f (x) � x 4 � 8 x 2 � 4. Write a function g (x) that performs each transformation.

A. Reflect f (x) across the x-axis.

When you reflect a function across the x-axis you take the of f (x).

g (x) � �f (x)

g (x) � ( x 4 � 8 x 2 � 4)

g (x) � � x 4 8 x 2 4

Check: Graph both functions on a graphing calculator. Do your graphs

appear to be reflections?

B. Reflect f (x) across the y-axis.

When you reflect a function across the y-axis you take the opposite of x, f (�x).

g (x) � f (�x)

g (x) � � (�x ) 4 � 8 � � 2 � 4 � g (x) � x 4 8 x 2 4

What do you notice about both functions?

Compressing and Stretching Polynomial FunctionsLet f (x) � x 4 � 6 x 2 � 3. Graph f and g on the same coordinate plane. Describe g as a transformation of f.

A. g (x) � 2f(x)

g (x) � ( x 4 � 6 x 2 � 3) Multiply each term by 2.

g (x) � x 4 � x 2 � 6

g (x) is a stretch of f (x).

f (x) is graphed on the grid. Graph g (x) on the same grid.

B. g (x) � f � 1 __ 2

x � g (x) � � 1 __ 2 x � � 6 � x � � 3

g (x) � x 4 � x 2 � 3

g (x) is a horizontal of f (x).

f (x) is graphed on the grid. Graph g (x) on the same grid.

x

y

4

8

–8

–4–2–4–6 2 4

x

y

4

8

–8

–4–2–4 2 42–– 22

–

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Ready To Go On? Skills Intervention6-9 Curve Fitting with Polynomial Models6B

SECTION

Using Finite Differences to Determine DegreesUse finite differences to determine the degree of the polynomial that best describes the data.

A. x �2 �1 0 1 2 3

y �47 �8 5 10 25 68

The x-values increase by a constant . Find the differences of the y-values.

To find the differences in the y-value subtract each y-value from the y-value that follows it. For example: �8 �(�47) � 39

y �47 �8 5 10 25 68

First differences: 39 13 Are the differences constant?

Second differences: �26 Are the differences constant?

Third differences: 18 Are the differences constant?

The differences are constant. Use the table above to determine the type of

polynomial that best describes the data. A polynomial best describes the data.

Use the cubic regression feature on your calculator to determine the polynomial.

f (x) � x � x 2 � x �

B. x �2 �1 0 1 2 3

y 51 10 5 6 7 26

The x-values increase by a constant . Find the differences of the y-values.

y 51 10 5 6 7 26

First differences: �41 �5 Are the differences constant?

Second differences: 36 Are the differences constant?

Third differences: �30 Are the differences constant?

Fourth differences: Are the differences constant?

The differences are constant. Use the table above to determine the type of

polynomial that best describes the data. A polynomial best describes the data.

Use the regression feature on your calculator to determine the polynomial.

f (x) � x � x � x 2 � x �

Finite Differences of Polynomials

Function Type

Degree Constant Finite Difference

Linear 1 First

Quadratic 2 Second

Cubic 3 Third

Quartic 4 Fourth

Quintic 5 Fifth

Name Date Class


To create a mathematical model for the given data, you will need to determine what type of function is most appropriate. Often, real-world data can be too irregular for you to use finite differences or find a polynomial function that fits perfectly. In these situations, you can use the regression feature on your graphing calculator.

The table shows the population of wild turkeys released into the wild at a state forest. Write a polynomial function for the population.

Time (years) 1 2 3 4 5

Number of Wild Turkeys

75 66 71 79 91

Understand the Problem

1. What are you being asked to do?

Make a Plan

2. In order to find the of the polynomial function, make a scatter plot.

Solve

3. Make a scatter plot of the data. Let x represent the number of years since the release.

4. Use the regression feature on your graphing calculator to check the R 2 values.

Quadratic R 2 �

Cubic R 2 �

Quartic R 2 �

Remember that the closer the R 2 -value is to 1, the better the function fits the data.

5. Which function is the most appropriate?

6. Write the polynomial model: f (x) �

Look Back

7. Check one of the points in the model in Exercise 6. Try x � 2.

f (2) � 0.5 4 � 6.8 3 � 35.5 2 � 75.2 � 121

f (2) �

8. Did you get a value near 66?

Ready To Go On? Problem Solving Intervention6-9 Curve Fitting with Polynomial Models6B

SECTION

y

x

100

80

60

40

20

2 4 6 8 100

6-5 Finding Real Roots of Polynomial Equations

1. The yearly profit of a company, in thousands of dollars, can be modeled by P(t ) � t 4 � 13 t 2 � 36, where t is the number of years since 2002. Factor to find the years in which the profit was 0.

Identify the roots of each equation. State the multiplicity of each root.

2. x 3 � 9 x 2 � 27x � 27 � 0 3. 3 x 3 � 3 x 2 � 48x � 60 � 0 4. x 4 � 8 x 3 � 16 x 2 � 0

6-6 Fundamental Theorem of AlgebraWrite the simplest polynomial function with the given roots.

5. �1, 2, 4

6. 2i, �2, 1

7. Solve x 4 � 2 x 3 � 4 x 2 � 8x � 32 � 0 by finding all roots.

6-7 Investigating Graphs of Polynomial FunctionsGraph each function.

8. f (x) � x 4 � 20 x 2 � 64 9. f (x) � x 3 � 3 x 2 � 6x � 8

y

x

–30

60

30

90

–2–4 2 4

y

x

–12

–6

6

12

–2–4 2 4


Name Date Class

Ready To Go On? Quiz6B

SECTION

Name Date Class


Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

10. 11. 12.

y

x

y

x

y

x

6-8 Transforming Polynomial FunctionsLet f (x) � x 4 � 2 x 2 � 5. Write a function g (x) that performs each transformation.

13. Reflect f (x) across the x-axis. 14. Reflect f (x) across the y-axis.

Let f (x) � 3 x 4 � 2 x 2 � 1. Graph f (x) and g (x) on the same coordinate plane. Describe g (x) as a transformation of f (x).

15. g (x) � 2f (x) 16. g (x) � 1 __ 4 f (x) 17. g (x) � f (x � 2)

y

x

–4

–2

2

4

–2–4 2 4

y

x

–4

–2

2

4

–2–4 2 4

y

x

–4

–2

2

4

–2–4 2 4

6-9 Curve Fitting with Polynomial Models

18. The table shows the population of a bacteria Time (h) 1 2 3 4 5

Number of bacteria

88 224 504 1030 1898colony over time. Write a polynomial function

for the data.

Ready To Go On? Quiz continued

6BSECTION

PackagingA box with a lid is to be made by cutting a 24-inch by 36-inch piece of cardboard along the dotted lines as shown in the figure and folding the flaps up along the sides. What is the maximum volume of the box? What value of x will produce a box of maximum value? What are the dimensions of the box?

x xx

xx

x

xx

24

36

1. Determine the dimensions of the box.

2. Write an equation for the volume of the box.

3. Use the Table feature on a graphing calculator with �Tbl � 1 to fill in the chart, where Y1 is the volume.

4. For what interval of x does the maximum volume appear to exist?

5. Reset the table by setting TblStart to the lower value of the interval in Exercise 4 and �Tbl to 0.1. In what interval of x does the maximum

volume appear to exist?

6. Reset the table by setting TblStart to the lower value of the interval in Exercise 5 and �Tbl to 0.05. What appears to be the value of x that

produces the maximum volume?

7. What is the maximum volume of the box?

8. What are the dimensions of the box?


Name Date Class

Ready To Go On? Enrichment6B

SECTION

X Y 1

1

2

3

4

5

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ready to go on? skills intervention 6a 6-1 polynomialsready to go on? skills intervention 6-1...

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