To determine The polynomial P ( x ) of degree 3 with real coefficients having zeros 4 and 1 + i such that it satisfies the condition P ( 2 ) = 4 . Explanation Given: The value of P ( 2 ) is 4 and the zeros of the polynomial P ( x ) are 4 and 1 + i . Theorems used: Conjugate zeros theorem: If P ( x ) is a polynomial function having only real coefficients, and if a + b i is a zeros of P ( x ) , then the conjugate a − b i is also a zero of P ( x ) . Factor theorem: If a polynomial P ( x ) has a zero k then the factor of that polynomial is ( x − k ) . Calculation: By conjugate zeroes theorem, if 1 + i is a zero of the polynomial then the conjugate term 1 − i must also be zero of that polynomial. The polynomial P ( x ) have three zeros 4 , 1 + i and 1 − i then by the factor theorem the linear factors of the polynomial are ( x − 4 ) , ( x − ( 1 + i ) ) and ( x − ( 1 − i ) ) . Therefore, the polynomial is written as P ( x ) = ( x − 4 ) ( x − ( 1 + i ) ) . Simplify the above polynomial by distributive property. P ( x ) = ( x − 4 ) ( x − ( 1 + i ) ) ( x − ( 1 − i ) ) = ( x − 4 ) ( x − 1 − i ) ( x − 1 + i ) Further, solve the above obtained polynomial expression. P ( x ) = ( x − 4 ) ( x − 1 − i ) ( x − 1 + i ) = ( x − 4 ) ( x 2 − 2 x + 2 ) = x 3 − 6 x 2 + 10 x − 8 Therefore, the value of polynomial P ( x ) is x 3 − 6 x 2 + 10 x − 8

7th Edition

ISBN: 9780134674377

Author: Rockswold

Publisher: VST

See similar textbooks

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Videos

Question

Chapter 4.3, Problem 11E

To determine

The polynomial $P(x)$ of degree $3$ with real coefficients having zeros $4$ and $1+i$ such that it satisfies the condition $P(2)=4$ .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.3, Problem 10E

Chapter 4.3, Problem 12E

Students have asked these similar questions

[2 01 3. (12 pts) Let A = 1 1 and b = 1 L2 1] 213 [2] (1) (2) Use the Gram-Schmidt process to find an orthonormal basis for the column space of A. Factor A into a product QR, where Q has an orthonormal set of column vectors and R is upper triangular. (3) Solve the least squares problem Ax = b.

22.) Mr. Vedrani has a pool with a deck built around it. The equation (2x+14)(2x+19) = 650 represents the relationship of the side lengths (in feet) of the entire pool area (in square feet). a.) If 350 represents the area of the pool and deck area, what do the expressions (2x+14) and (2x+19) each represent? b.) Using graphing technology, find x, the width of the deck around the pool. If necessary, round to the nearest tenth. x=

How do I get common factors?

Chapter 4 Solutions

EBK GRAPHICAL APPROACH TO ALGEBRA & TRI

Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Use an end behavior diagram , , , or to describe...Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Without using a calculator, match each function in...Ch. 4.1 - Prob. 34E Ch. 4.1 - Without using a calculator, match each function in...Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - CONCEPT CHECK The graphs below show but not...Ch. 4.1 - Prob. 46E Ch. 4.1 - CONCEPT CHECK The graphs below show y = x3 − 3x2 −...Ch. 4.1 - Prob. 48E Ch. 4.1 - CONCEPT CHECK The graphs below show but not...Ch. 4.1 - Prob. 50E Ch. 4.1 - CONCEPT CHECK The graphs below show y = x3 − 3x2 −...Ch. 4.1 - Prob. 52E Ch. 4.1 - CONCEPT CHECK The graphs below show y = x3 − 3x2 −...Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - Prob. 57E Ch. 4.1 - Prob. 58E Ch. 4.1 - Prob. 59E Ch. 4.1 - Prob. 60E Ch. 4.1 - Prob. 61E Ch. 4.1 - Prob. 62E Ch. 4.1 - Prob. 63E Ch. 4.1 - Prob. 64E Ch. 4.1 - Prob. 65E Ch. 4.1 - Prob. 66E Ch. 4.1 - Prob. 67E Ch. 4.1 - Prob. 68E Ch. 4.1 - Prob. 69E Ch. 4.1 - Prob. 70E Ch. 4.1 - Prob. 71E Ch. 4.1 - Prob. 72E Ch. 4.1 - Prob. 73E Ch. 4.1 - Prob. 74E Ch. 4.2 - CHECKING ANALYTIC SKILLS Simplify each rational...Ch. 4.2 - CHECKING ANALYTIC SKILLS Simplify each rational...Ch. 4.2 - Prob. 3E Ch. 4.2 - Prob. 4E Ch. 4.2 - CHECKING ANALYTIC SKILLS Simplify each rational...Ch. 4.2 - Prob. 6E Ch. 4.2 - Prob. 7E Ch. 4.2 - Prob. 8E Ch. 4.2 - Use the intermediate value theorem to show that...Ch. 4.2 - Prob. 10E Ch. 4.2 - Prob. 11E Ch. 4.2 - Prob. 12E Ch. 4.2 - Use the intermediate value theorem to show that...Ch. 4.2 - Prob. 14E Ch. 4.2 - Prob. 15E Ch. 4.2 - Prob. 16E Ch. 4.2 - Use the intermediate value theorem to show that...Ch. 4.2 - Prob. 18E Ch. 4.2 - CONCEPT CHECK Suppose that a polynomial function P...Ch. 4.2 - Prob. 20E Ch. 4.2 - Find each quotient when P(x) is divided by the...Ch. 4.2 - Prob. 22E Ch. 4.2 - Prob. 23E Ch. 4.2 - Prob. 24E Ch. 4.2 - Find each quotient when P(x) is divided by the...Ch. 4.2 - Prob. 26E Ch. 4.2 - Prob. 27E Ch. 4.2 - Prob. 28E Ch. 4.2 - Find each quotient when P(x) is divided by the...Ch. 4.2 - Prob. 30E Ch. 4.2 - Prob. 31E Ch. 4.2 - Prob. 32E Ch. 4.2 - Use synthetic substitution to find P(k). k = 3;...Ch. 4.2 - Prob. 34E Ch. 4.2 - Prob. 35E Ch. 4.2 - Prob. 36E Ch. 4.2 - Use synthetic substitution to find P(k). k = 2;...Ch. 4.2 - Prob. 38E Ch. 4.2 - Prob. 39E Ch. 4.2 - Prob. 40E Ch. 4.2 - Use synthetic substitution to find P(k). ; P(x) =...Ch. 4.2 - Prob. 42E Ch. 4.2 - Prob. 43E Ch. 4.2 - Prob. 44E Ch. 4.2 - Use synthetic substitution to determine whether...Ch. 4.2 - Prob. 46E Ch. 4.2 - Prob. 47E Ch. 4.2 - Prob. 48E Ch. 4.2 - Use synthetic substitution to determine whether...Ch. 4.2 - Prob. 50E Ch. 4.2 - Prob. 51E Ch. 4.2 - Prob. 52E Ch. 4.2 - Use synthetic substitution to determine whether...Ch. 4.2 - Prob. 54E Ch. 4.2 - Prob. 55E Ch. 4.2 - Prob. 56E Ch. 4.2 - Prob. 57E Ch. 4.2 - Prob. 58E Ch. 4.2 - Prob. 59E Ch. 4.2 - Prob. 60E Ch. 4.2 - For each polynomial at least one zero is given....Ch. 4.2 - Prob. 62E Ch. 4.2 - For each polynomial at least one zero is given....Ch. 4.2 - Prob. 64E Ch. 4.2 - For each polynomial at least one zero is given....Ch. 4.2 - For each polynomial at least one zero is given....Ch. 4.2 - Prob. 67E Ch. 4.2 - Prob. 68E Ch. 4.2 - For each polynomial at least one zero is given....Ch. 4.2 - Prob. 70E Ch. 4.2 - Prob. 71E Ch. 4.2 - Prob. 72E Ch. 4.2 - Factor P(x) into linear factors given that k is a...Ch. 4.2 - Prob. 74E Ch. 4.2 - Prob. 75E Ch. 4.2 - Prob. 76E Ch. 4.2 - Factor P(x) into linear factors given that k is a...Ch. 4.2 - Prob. 78E Ch. 4.2 - Prob. 79E Ch. 4.2 - Prob. 80E Ch. 4.2 - Divide. Ch. 4.2 - Prob. 82E Ch. 4.2 - Divide. Ch. 4.2 - Prob. 84E Ch. 4.2 - Divide. Ch. 4.2 - Prob. 86E Ch. 4.2 - Prob. 87E Ch. 4.2 - Prob. 88E Ch. 4.2 - Prob. 89E Ch. 4.2 - Prob. 90E Ch. 4.2 - Prob. 91E Ch. 4.2 - Prob. 92E Ch. 4.2 - Prob. 1RBC Ch. 4.2 - Prob. 2RBC Ch. 4.2 - Prob. 3RBC Ch. 4.2 - Prob. 4RBC Ch. 4.2 - Prob. 5RBC Ch. 4.2 - Prob. 6RBC Ch. 4.2 - Prob. 7RBC Ch. 4.2 - Prob. 8RBC Ch. 4.2 - Prob. 9RBC Ch. 4.2 - Prob. 10RBC Ch. 4.3 - CHECKING ANALYTIC SKILLS Find a cubic polynomial...Ch. 4.3 - Prob. 2E Ch. 4.3 - CHECKING ANALYTIC SKILLS Find a cubic polynomial...Ch. 4.3 - Prob. 4E Ch. 4.3 - CHECKING ANALYTIC SKILLS Find a cubic polynomial...Ch. 4.3 - CHECKING ANALYTIC SKILLS Find a cubic polynomial...Ch. 4.3 - CHECKING ANALYTIC SKILLS Find a polynomial...Ch. 4.3 - Prob. 8E Ch. 4.3 - CHECKING ANALYTIC SKILLS Find a polynomial...Ch. 4.3 - Prob. 10E Ch. 4.3 - CHECKING ANALYTIC SKILLS Find a polynomial...Ch. 4.3 - Prob. 12E Ch. 4.3 - One or more zeros are given for each polynomial....Ch. 4.3 - Prob. 14E Ch. 4.3 - One or more zeros are given for each polynomial....Ch. 4.3 - Prob. 16E Ch. 4.3 - One or more zeros are given for each polynomial....Ch. 4.3 - Prob. 18E Ch. 4.3 - Find a polynomial function P(x) having leading...Ch. 4.3 - Prob. 20E Ch. 4.3 - Find a polynomial function P(x) having leading...Ch. 4.3 - Prob. 22E Ch. 4.3 - Find a polynomial function P(x) having leading...Ch. 4.3 - Prob. 24E Ch. 4.3 - Prob. 25E Ch. 4.3 - Prob. 26E Ch. 4.3 - Find a polynomial function P(x) having leading...Ch. 4.3 - Prob. 28E Ch. 4.3 - Prob. 29E Ch. 4.3 - Prob. 30E Ch. 4.3 - Sketch by hand the graph of each function. (You...Ch. 4.3 - Prob. 32E Ch. 4.3 - Sketch by hand the graph of each function. (You...Ch. 4.3 - Prob. 34E Ch. 4.3 - Sketch by hand the graph of each function. (You...Ch. 4.3 - Prob. 36E Ch. 4.3 - Sketch by hand the graph of each function. (You...Ch. 4.3 - Prob. 38E Ch. 4.3 - Prob. 39E Ch. 4.3 - Prob. 40E Ch. 4.3 - CONCEPT CHECK Use the graphs in Exercises 41–46 to...Ch. 4.3 - CONCEPT CHECK Use the graphs in Exercises 41–46 to...Ch. 4.3 - CONCEPT CHECK Use the graphs in Exercises 41–46 to...Ch. 4.3 - Prob. 44E Ch. 4.3 - CONCEPT CHECK Use the graphs in Exercises 41–46 to...Ch. 4.3 - Prob. 46E Ch. 4.3 - Prob. 47E Ch. 4.3 - Prob. 48E Ch. 4.3 - Prob. 49E Ch. 4.3 - Prob. 50E Ch. 4.3 - Prob. 51E Ch. 4.3 - Prob. 52E Ch. 4.3 - For each polynomial function, (a) list all...Ch. 4.3 - Prob. 54E Ch. 4.3 - For each polynomial function, (a) list all...Ch. 4.3 - Prob. 56E Ch. 4.3 - For each polynomial function, (a) list all...Ch. 4.3 - Prob. 58E Ch. 4.3 - Prob. 59E Ch. 4.3 - Prob. 60E Ch. 4.3 - Prob. 61E Ch. 4.3 - Prob. 62E Ch. 4.3 - Prob. 63E Ch. 4.3 - Prob. 64E Ch. 4.3 - Prob. 65E Ch. 4.3 - Prob. 66E Ch. 4.3 - Prob. 67E Ch. 4.3 - Prob. 68E Ch. 4.3 - Prob. 69E Ch. 4.3 - Prob. 70E Ch. 4.3 - Prob. 71E Ch. 4.3 - Prob. 72E Ch. 4.3 - Prob. 73E Ch. 4.3 - Prob. 74E Ch. 4.3 - Prob. 75E Ch. 4.3 - Prob. 76E Ch. 4.3 - Prob. 77E Ch. 4.3 - Prob. 78E Ch. 4.3 - Prob. 79E Ch. 4.3 - Prob. 80E Ch. 4.3 - Prob. 81E Ch. 4.3 - Prob. 82E Ch. 4.3 - Prob. 83E Ch. 4.3 - Prob. 84E Ch. 4.3 - Prob. 85E Ch. 4.3 - Prob. 86E Ch. 4.3 - Prob. 87E Ch. 4.3 - Prob. 88E Ch. 4.3 - Prob. 89E Ch. 4.3 - Prob. 90E Ch. 4.3 - Prob. 91E Ch. 4.3 - Prob. 92E Ch. 4.3 - RELATING CONCEPTS For individual or group...Ch. 4.3 - Prob. 94E Ch. 4.3 - Prob. 95E Ch. 4.3 - do the following in order. Use Descartes’ rule of...Ch. 4.3 - Prob. 97E Ch. 4.3 - Prob. 98E Ch. 4.4 - Prob. 1E Ch. 4.4 - Prob. 2E Ch. 4.4 - Prob. 3E Ch. 4.4 - Prob. 4E Ch. 4.4 - Prob. 5E Ch. 4.4 - Prob. 6E Ch. 4.4 - Prob. 7E Ch. 4.4 - Prob. 8E Ch. 4.4 - CHECKING ANALYTIC SKILLS Find all real solutions....Ch. 4.4 - Prob. 10E Ch. 4.4 - Prob. 11E Ch. 4.4 - Prob. 12E Ch. 4.4 - CHECKING ANALYTIC SKILLS Find all complex...Ch. 4.4 - Prob. 14E Ch. 4.4 - Prob. 15E Ch. 4.4 - Prob. 16E Ch. 4.4 - CHECKING ANALYTIC SKILLS Find all complex...Ch. 4.4 - Prob. 18E Ch. 4.4 - Prob. 19E Ch. 4.4 - Solve each equation analytically for all complex...Ch. 4.4 - Solve each equation analytically for all complex...Ch. 4.4 - Prob. 22E Ch. 4.4 - Prob. 23E Ch. 4.4 - Prob. 24E Ch. 4.4 - Prob. 25E Ch. 4.4 - Prob. 26E Ch. 4.4 - Prob. 27E Ch. 4.4 - Prob. 28E Ch. 4.4 - Prob. 29E Ch. 4.4 - Prob. 30E Ch. 4.4 - Prob. 31E Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Prob. 34E Ch. 4.4 - Prob. 35E Ch. 4.4 - Prob. 36E Ch. 4.4 - Graph each polynomial function by hand, as shown...Ch. 4.4 - Prob. 38E Ch. 4.4 - Graph each polynomial function by hand, as shown...Ch. 4.4 - Prob. 40E Ch. 4.4 - Graph each polynomial function by hand, as shown...Ch. 4.4 - Prob. 42E Ch. 4.4 - Prob. 43E Ch. 4.4 - Prob. 44E Ch. 4.4 - Prob. 45E Ch. 4.4 - Prob. 46E Ch. 4.4 - Prob. 47E Ch. 4.4 - Prob. 48E Ch. 4.4 - Prob. 49E Ch. 4.4 - Prob. 50E Ch. 4.4 - Prob. 51E Ch. 4.4 - Prob. 52E Ch. 4.4 - Prob. 53E Ch. 4.4 - Prob. 54E Ch. 4.4 - Find all n complex solutions of each equation of...Ch. 4.4 - Prob. 56E Ch. 4.4 - Prob. 57E Ch. 4.4 - Prob. 58E Ch. 4.4 - Find all n complex solutions of each equation of...Ch. 4.4 - Prob. 60E Ch. 4.4 - Prob. 61E Ch. 4.4 - Prob. 62E Ch. 4.4 - Find all n complex solutions of each equation of...Ch. 4.4 - Prob. 64E Ch. 4.4 - Prob. 65E Ch. 4.4 - Prob. 66E Ch. 4.4 - Prob. 67E Ch. 4.4 - Prob. 68E Ch. 4.4 - Prob. 69E Ch. 4.4 - Prob. 70E Ch. 4.4 - Prob. 71E Ch. 4.4 - Prob. 72E Ch. 4.4 - Prob. 73E Ch. 4.4 - Prob. 74E Ch. 4.4 - Prob. 75E Ch. 4.4 - Prob. 76E Ch. 4.4 - Prob. 77E Ch. 4.4 - Prob. 78E Ch. 4.4 - Prob. 1RBC Ch. 4.4 - Prob. 2RBC Ch. 4.4 - Prob. 3RBC Ch. 4.4 - Prob. 4RBC Ch. 4.4 - Prob. 5RBC Ch. 4.4 - Prob. 6RBC Ch. 4.4 - Prob. 7RBC Ch. 4.4 - Prob. 8RBC Ch. 4 - Prob. 1RE Ch. 4 - Prob. 2RE Ch. 4 - Prob. 3RE Ch. 4 - Prob. 4RE Ch. 4 - Prob. 5RE Ch. 4 - Prob. 6RE Ch. 4 - Prob. 7RE Ch. 4 - Prob. 8RE Ch. 4 - Prob. 9RE Ch. 4 - Prob. 10RE Ch. 4 - Prob. 11RE Ch. 4 - Prob. 12RE Ch. 4 - Prob. 13RE Ch. 4 - Prob. 14RE Ch. 4 - Prob. 15RE Ch. 4 - Prob. 16RE Ch. 4 - Prob. 17RE Ch. 4 - Prob. 18RE Ch. 4 - Prob. 19RE Ch. 4 - Prob. 20RE Ch. 4 - Prob. 21RE Ch. 4 - Prob. 22RE Ch. 4 - Prob. 23RE Ch. 4 - Prob. 24RE Ch. 4 - Prob. 25RE Ch. 4 - Prob. 26RE Ch. 4 - Prob. 27RE Ch. 4 - Prob. 28RE Ch. 4 - Give the solution set of each inequality, using...Ch. 4 - Prob. 30RE Ch. 4 - Prob. 31RE Ch. 4 - Prob. 32RE Ch. 4 - Prob. 33RE Ch. 4 - Prob. 34RE Ch. 4 - Prob. 35RE Ch. 4 - Prob. 36RE Ch. 4 - Prob. 37RE Ch. 4 - Prob. 38RE Ch. 4 - Prob. 39RE Ch. 4 - Prob. 40RE Ch. 4 - Prob. 41RE Ch. 4 - Prob. 42RE Ch. 4 - Prob. 43RE Ch. 4 - Prob. 44RE Ch. 4 - Prob. 45RE Ch. 4 - Prob. 46RE Ch. 4 - Prob. 47RE Ch. 4 - Prob. 48RE Ch. 4 - Prob. 49RE Ch. 4 - Prob. 50RE Ch. 4 - Prob. 51RE Ch. 4 - Prob. 52RE Ch. 4 - Prob. 53RE Ch. 4 - Prob. 54RE Ch. 4 - Prob. 55RE Ch. 4 - Prob. 56RE Ch. 4 - Prob. 1T Ch. 4 - Prob. 2T Ch. 4 - Consider the function Use a calculator to give a...Ch. 4 - Prob. 4T Ch. 4 - Prob. 5T Ch. 4 - Prob. 6T Ch. 4 - Prob. 7T Ch. 4 - Prob. 8T Ch. 4 - Prob. 9T Ch. 4 - Prob. 10T Ch. 4 - Prob. 11T Ch. 4 - Prob. 12T Ch. 4 - Prob. 13T Ch. 4 - Prob. 14T

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.