Precalculus with Limits: A Graphing Approach
Precalculus with Limits: A Graphing Approach
7th Edition
ISBN: 9781305071711
Author: Ron Larson
Publisher: Brooks/Cole
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The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form o…
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Chapter 2.3, Problem 54E

a.

To determine

To verify whether (x+3) is a factor to the function f(x)=3x3+2x2−19x+6 .

a.

Expert Solution
Check Mark

Answer to Problem 54E

Factor

Explanation of Solution

Given:

Function: f(x)=3x3+2x2−19x+6

Factor: x+3

Calculation:

Verifying whether x+3 is a factor to given function or not by using synthetic division method.

Step 1: Writing coefficients of polynomial function on the right side and value of constant of factor with opposite sign on the left side.

  −332−196

Step 2: Now, adding terms in columns and multiplying the results by -3.

  −332−196−921−63−720 ←Remainder

Here, the remainder is zero, it means that x+3 is a factor to given polynomial function.

Conclusion:

Therefore, x+3 is a factor to given polynomial function.

b.

To determine

Find the remaining factors of the function f(x)=3x3+2x2−19x+6 .

b.

Expert Solution
Check Mark

Answer to Problem 54E

  (x−2),(3x−1)

Explanation of Solution

Given:

Function: f(x)=3x3+2x2−19x+6

Calculation:

From above answer, the quotient is 3x2−7x+2 .

Hence, f(x)=3x3+2x2−19x+6    ⇔    f(x)=(x+3)(3x2−7x+2)

  ⇒f(x)=(x+3)(3x2−7x+2)⇒f(x)=(x+3)(3x2−6x−x+2)⇒f(x)=(x+3)(3x(x−2)−1(x−2))⇒f(x)=(x+3)(x−2)(3x−1)

Conclusion:

Therefore, (x−2),(3x−1) are the remaining factors to the given polynomial function.

c.

To determine

Write the complete factorization of function f(x)=3x3+2x2−19x+6 .

c.

Expert Solution
Check Mark

Answer to Problem 54E

  f(x)=(x+3)(x−2)(3x−1)

Explanation of Solution

Given:

Function: f(x)=3x3+2x2−19x+6

Calculation:

From above answers, the complete factorization of given polynomial function is

  f(x)=(x+3)(x−2)(3x−1) .

Conclusion:

Therefore, f(x)=(x+3)(x−2)(3x−1) is complete factorization to given polynomial function.

d.

To determine

Find the zeros of the function f(x)=3x3+2x2−19x+6 .

d.

Expert Solution
Check Mark

Answer to Problem 54E

  −3,13,2

Explanation of Solution

Given:

Function: f(x)=3x3+2x2−19x+6

Calculation:

To find the zeros, put f(x)=0.

  ⇒(x+3)(x−2)(3x−1)=0⇒x=−3,13,2

Calculation for graph:

Consider f(x)=3x3+2x2−19x+6

    Values of xValues of f (x)
    06
    1-8
    -124
    20
    -228

By taking different values of x, the graph can be plotted.

Graph:

Precalculus with Limits: A Graphing Approach, Chapter 2.3, Problem 54E

Interpretation:

By observing graph, it is clear that the curve of the function meets x-axis at x=−3,13 and 2.

Hence, the zeros of the function are x=−3,13 and 2.

Conclusion:

Therefore, the zeros of given polynomial function are x=−3,13 and 2.

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Chapter 2.3, Problem 53E
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Chapter 2.3, Problem 55E

Chapter 2 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 2.1 - Prob. 1ECh. 2.1 - Prob. 2ECh. 2.1 - Prob. 3ECh. 2.1 - Prob. 4ECh. 2.1 - Prob. 5ECh. 2.1 - Prob. 6ECh. 2.1 - Prob. 7ECh. 2.1 - Prob. 8ECh. 2.1 - Prob. 9ECh. 2.1 - Prob. 10E
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Prob. 20ECh. 2.6 - Prob. 21ECh. 2.6 - Prob. 22ECh. 2.6 - Prob. 23ECh. 2.6 - Prob. 24ECh. 2.6 - Prob. 25ECh. 2.6 - Prob. 26ECh. 2.6 - Prob. 27ECh. 2.6 - Prob. 28ECh. 2.6 - Prob. 29ECh. 2.6 - Prob. 30ECh. 2.6 - Prob. 31ECh. 2.6 - Prob. 32ECh. 2.6 - Prob. 33ECh. 2.6 - Prob. 34ECh. 2.6 - Prob. 35ECh. 2.6 - Prob. 36ECh. 2.6 - Prob. 37ECh. 2.6 - Prob. 38ECh. 2.6 - Prob. 39ECh. 2.6 - Prob. 40ECh. 2.6 - Prob. 41ECh. 2.6 - Prob. 42ECh. 2.6 - Prob. 43ECh. 2.6 - Prob. 44ECh. 2.6 - Prob. 45ECh. 2.6 - Prob. 46ECh. 2.6 - Prob. 47ECh. 2.6 - Prob. 48ECh. 2.6 - Prob. 49ECh. 2.6 - Prob. 50ECh. 2.6 - Prob. 51ECh. 2.6 - Prob. 52ECh. 2.6 - Prob. 53ECh. 2.6 - Prob. 54ECh. 2.6 - Prob. 55ECh. 2.6 - Prob. 56ECh. 2.6 - Prob. 57ECh. 2.6 - Prob. 58ECh. 2.6 - Prob. 59ECh. 2.6 - Prob. 60ECh. 2.6 - Prob. 61ECh. 2.6 - Prob. 62ECh. 2.6 - Prob. 63ECh. 2.6 - Prob. 64ECh. 2.7 - Prob. 1ECh. 2.7 - Prob. 2ECh. 2.7 - Prob. 3ECh. 2.7 - Prob. 4ECh. 2.7 - Prob. 5ECh. 2.7 - Prob. 6ECh. 2.7 - Prob. 7ECh. 2.7 - Prob. 8ECh. 2.7 - Prob. 9ECh. 2.7 - Prob. 10ECh. 2.7 - Prob. 11ECh. 2.7 - Prob. 12ECh. 2.7 - Prob. 13ECh. 2.7 - Prob. 14ECh. 2.7 - Prob. 15ECh. 2.7 - Prob. 16ECh. 2.7 - Prob. 17ECh. 2.7 - Prob. 18ECh. 2.7 - Prob. 19ECh. 2.7 - Prob. 20ECh. 2.7 - Prob. 21ECh. 2.7 - Prob. 22ECh. 2.7 - Prob. 23ECh. 2.7 - Prob. 24ECh. 2.7 - Prob. 25ECh. 2.7 - Prob. 26ECh. 2.7 - Prob. 27ECh. 2.7 - Prob. 28ECh. 2.7 - Prob. 29ECh. 2.7 - Prob. 30ECh. 2.7 - Prob. 31ECh. 2.7 - Prob. 32ECh. 2.7 - Prob. 33ECh. 2.7 - Prob. 34ECh. 2.7 - Prob. 35ECh. 2.7 - Prob. 36ECh. 2.7 - Prob. 37ECh. 2.7 - Prob. 38ECh. 2.7 - Prob. 39ECh. 2.7 - Prob. 40ECh. 2.7 - Prob. 41ECh. 2.7 - Prob. 42ECh. 2.7 - Prob. 43ECh. 2.7 - Prob. 44ECh. 2.7 - Prob. 45ECh. 2.7 - Prob. 46ECh. 2.7 - Prob. 47ECh. 2.7 - Prob. 48ECh. 2.7 - Prob. 49ECh. 2.7 - Prob. 50ECh. 2.7 - Prob. 51ECh. 2.7 - Prob. 52ECh. 2.7 - Prob. 53ECh. 2.7 - Prob. 54ECh. 2.7 - Prob. 55ECh. 2.7 - Prob. 56ECh. 2.7 - Prob. 57ECh. 2.7 - Prob. 58ECh. 2.7 - Prob. 59ECh. 2.7 - Prob. 60ECh. 2.7 - Prob. 61ECh. 2.7 - Prob. 62ECh. 2.7 - Prob. 63ECh. 2.7 - Prob. 64ECh. 2.7 - Prob. 65ECh. 2.7 - Prob. 66ECh. 2.7 - Prob. 67ECh. 2.7 - Prob. 68ECh. 2.7 - Prob. 69ECh. 2.7 - Prob. 70ECh. 2.7 - Prob. 71ECh. 2.7 - Prob. 72ECh. 2.7 - Prob. 73ECh. 2.7 - Prob. 74ECh. 2.7 - Prob. 75ECh. 2.7 - Prob. 76ECh. 2.7 - Prob. 77ECh. 2.7 - Prob. 78ECh. 2.7 - Prob. 79ECh. 2.7 - Prob. 80ECh. 2.7 - Prob. 81ECh. 2.7 - Prob. 82ECh. 2.7 - Prob. 83ECh. 2.7 - Prob. 84ECh. 2.7 - Prob. 85ECh. 2.7 - Prob. 86ECh. 2.7 - Prob. 87ECh. 2.7 - Prob. 88ECh. 2.7 - Prob. 89ECh. 2.7 - Prob. 90ECh. 2.7 - Prob. 91ECh. 2.7 - Prob. 92ECh. 2.7 - Prob. 93ECh. 2.7 - Prob. 94ECh. 2.7 - Prob. 95ECh. 2.7 - Prob. 96ECh. 2.7 - Prob. 97ECh. 2.7 - Prob. 98ECh. 2.7 - Prob. 99ECh. 2.7 - Prob. 100ECh. 2.7 - Prob. 101ECh. 2.7 - Prob. 102ECh. 2.7 - Prob. 103ECh. 2.7 - Prob. 104ECh. 2.7 - Prob. 105ECh. 2.7 - Prob. 106ECh. 2.7 - Prob. 107ECh. 2.8 - Prob. 1ECh. 2.8 - Prob. 2ECh. 2.8 - Prob. 3ECh. 2.8 - Prob. 4ECh. 2.8 - Prob. 5ECh. 2.8 - Prob. 6ECh. 2.8 - Prob. 7ECh. 2.8 - Prob. 8ECh. 2.8 - Prob. 9ECh. 2.8 - Prob. 10ECh. 2.8 - Prob. 11ECh. 2.8 - Prob. 12ECh. 2.8 - Prob. 13ECh. 2.8 - Prob. 14ECh. 2.8 - Prob. 15ECh. 2.8 - Prob. 16ECh. 2.8 - Prob. 17ECh. 2.8 - Prob. 18ECh. 2.8 - Prob. 19ECh. 2.8 - Prob. 20ECh. 2.8 - Prob. 21ECh. 2.8 - Prob. 22ECh. 2.8 - Prob. 23ECh. 2.8 - Prob. 24ECh. 2.8 - Prob. 25ECh. 2.8 - Prob. 26ECh. 2.8 - Prob. 27ECh. 2.8 - Prob. 28ECh. 2.8 - Prob. 29ECh. 2.8 - Prob. 30ECh. 2.8 - Prob. 31ECh. 2.8 - Prob. 32ECh. 2.8 - Prob. 33ECh. 2.8 - Prob. 34ECh. 2.8 - Prob. 35ECh. 2.8 - Prob. 36ECh. 2.8 - Prob. 37ECh. 2.8 - Prob. 38ECh. 2 - Prob. 1RECh. 2 - Prob. 2RECh. 2 - Prob. 3RECh. 2 - Prob. 4RECh. 2 - Prob. 5RECh. 2 - Prob. 6RECh. 2 - Prob. 7RECh. 2 - Prob. 8RECh. 2 - Prob. 9RECh. 2 - Prob. 10RECh. 2 - Prob. 11RECh. 2 - Prob. 12RECh. 2 - Prob. 13RECh. 2 - Prob. 14RECh. 2 - Prob. 15RECh. 2 - Prob. 16RECh. 2 - Prob. 17RECh. 2 - Prob. 18RECh. 2 - Prob. 19RECh. 2 - Prob. 20RECh. 2 - Prob. 21RECh. 2 - Prob. 22RECh. 2 - Prob. 23RECh. 2 - Prob. 24RECh. 2 - Prob. 25RECh. 2 - Prob. 26RECh. 2 - Prob. 27RECh. 2 - Prob. 28RECh. 2 - Prob. 29RECh. 2 - Prob. 30RECh. 2 - Prob. 31RECh. 2 - Prob. 32RECh. 2 - Prob. 33RECh. 2 - Prob. 34RECh. 2 - Prob. 35RECh. 2 - Prob. 36RECh. 2 - Prob. 37RECh. 2 - Prob. 38RECh. 2 - Prob. 39RECh. 2 - Prob. 40RECh. 2 - Prob. 41RECh. 2 - Prob. 42RECh. 2 - Prob. 43RECh. 2 - Prob. 44RECh. 2 - Prob. 45RECh. 2 - Prob. 46RECh. 2 - Prob. 47RECh. 2 - Prob. 48RECh. 2 - Prob. 49RECh. 2 - Prob. 50RECh. 2 - Prob. 51RECh. 2 - Prob. 52RECh. 2 - Prob. 53RECh. 2 - Prob. 54RECh. 2 - Prob. 55RECh. 2 - Prob. 56RECh. 2 - Prob. 57RECh. 2 - Prob. 58RECh. 2 - Prob. 59RECh. 2 - Prob. 60RECh. 2 - Prob. 61RECh. 2 - Prob. 62RECh. 2 - Prob. 63RECh. 2 - Prob. 64RECh. 2 - Prob. 65RECh. 2 - Prob. 66RECh. 2 - Prob. 67RECh. 2 - Prob. 68RECh. 2 - Prob. 69RECh. 2 - Prob. 70RECh. 2 - Prob. 71RECh. 2 - Prob. 72RECh. 2 - Prob. 73RECh. 2 - Prob. 74RECh. 2 - Prob. 75RECh. 2 - Prob. 76RECh. 2 - Prob. 77RECh. 2 - Prob. 78RECh. 2 - Prob. 79RECh. 2 - Prob. 80RECh. 2 - Prob. 81RECh. 2 - Prob. 82RECh. 2 - Prob. 83RECh. 2 - Prob. 84RECh. 2 - Prob. 85RECh. 2 - Prob. 86RECh. 2 - Prob. 87RECh. 2 - Prob. 88RECh. 2 - Prob. 89RECh. 2 - Prob. 90RECh. 2 - Prob. 91RECh. 2 - Prob. 92RECh. 2 - Prob. 93RECh. 2 - Prob. 94RECh. 2 - Prob. 95RECh. 2 - Prob. 96RECh. 2 - Prob. 97RECh. 2 - Prob. 98RECh. 2 - Prob. 99RECh. 2 - Prob. 100RECh. 2 - Prob. 101RECh. 2 - Prob. 102RECh. 2 - Prob. 103RECh. 2 - Prob. 104RECh. 2 - Prob. 105RECh. 2 - Prob. 106RECh. 2 - Prob. 107RECh. 2 - Prob. 108RECh. 2 - Prob. 109RECh. 2 - Prob. 110RECh. 2 - Prob. 111RECh. 2 - Prob. 112RECh. 2 - Prob. 113RECh. 2 - Prob. 114RECh. 2 - Prob. 115RECh. 2 - Prob. 116RECh. 2 - Prob. 117RECh. 2 - Prob. 118RECh. 2 - Prob. 119RECh. 2 - Prob. 120RECh. 2 - Prob. 121RECh. 2 - Prob. 122RECh. 2 - Prob. 123RECh. 2 - Prob. 124RECh. 2 - Prob. 125RECh. 2 - Prob. 126RECh. 2 - Prob. 127RECh. 2 - Prob. 128RECh. 2 - Prob. 129RECh. 2 - Prob. 130RECh. 2 - Prob. 131RECh. 2 - Prob. 132RECh. 2 - Prob. 133RECh. 2 - Prob. 134RECh. 2 - Prob. 135RECh. 2 - Prob. 136RECh. 2 - Prob. 137RECh. 2 - Prob. 138RECh. 2 - Prob. 139RECh. 2 - Prob. 140RECh. 2 - Prob. 141RECh. 2 - Prob. 142RECh. 2 - Prob. 143RECh. 2 - Prob. 144RECh. 2 - Prob. 145RECh. 2 - Prob. 146RECh. 2 - Prob. 147RECh. 2 - Prob. 148RECh. 2 - Prob. 149RECh. 2 - Prob. 150RECh. 2 - Prob. 151RECh. 2 - Prob. 152RECh. 2 - Prob. 153RECh. 2 - Prob. 154RECh. 2 - Prob. 155RECh. 2 - Prob. 156RECh. 2 - Prob. 157RECh. 2 - Prob. 158RECh. 2 - Prob. 159RECh. 2 - Prob. 160RECh. 2 - Prob. 161RECh. 2 - Prob. 162RECh. 2 - Prob. 163RECh. 2 - Prob. 164RECh. 2 - Prob. 165RECh. 2 - Prob. 166RECh. 2 - Prob. 167RECh. 2 - Prob. 168RECh. 2 - Prob. 1CTCh. 2 - Prob. 2CTCh. 2 - Prob. 3CTCh. 2 - Prob. 4CTCh. 2 - Prob. 5CTCh. 2 - Prob. 6CTCh. 2 - Prob. 7CTCh. 2 - Prob. 8CTCh. 2 - Prob. 9CTCh. 2 - Prob. 10CTCh. 2 - Prob. 11CTCh. 2 - Prob. 12CTCh. 2 - Prob. 13CTCh. 2 - Prob. 14CTCh. 2 - Prob. 15CTCh. 2 - Prob. 16CTCh. 2 - Prob. 17CTCh. 2 - Prob. 18CTCh. 2 - Prob. 19CTCh. 2 - Prob. 20CTCh. 2 - Prob. 21CTCh. 2 - Prob. 22CTCh. 2 - Prob. 23CT
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