Answer (a), (b), and (c) only. A polynomial f(z) has the factor-square property (or FSP) if f(x) is a factor of f(r²).For instance, g(x) =1-1 and h(z) = r have FSP, but k(z) = 1+2 does not.Reason: z-1 is a factor of r-1, and r is a factor of r, but z+2 is not a factor of r +2.Multiplying by a nonzero constant “preserves" FSP, so we restrict attention to poly-nomials that are monic (ie., have 1 as highest-degree coefficient).What patterns do monic FSP polynomials satisfy?To make progress on this topic, investigate the following questions and justify youranswers.(a) Are z and r - 1 the only monic FSP polynomials of degree 1?(b) List all the monic FSP polynomials of degree 2.To start, note that r, r -1, r – 1, and r +1+1 are on that list.Some of them are products of FSP polynomials of smaller degree. For instance,7 and r-r arise from degree 1 cases. However, r -1 and r +r+1 are new,not expressible as a product of two smaller FSP polynomials.Which terms in your list of degree 2 examples are new?(c) List all the new monic FSP polynomials of degree 3.Note: Some monic FSP polynomials of degree 3 have complex coefficienta that are not real.Can you make a similar list in degree 4?(d) Are there monic FSP polynomials (of some degree) that have real numbercoefficients, but some of those coefficients are not integers?Explain your reasoning.

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Description: Answer (a), (b), and (c) only. A polynomial f(z) has the factor-square property (or FSP) if f(x) is a factor of f(r²).For instance, g(x) =1-1 and h(z) = r have FSP, but k(z) = 1+2 does not.Reason: z-1 is a factor of r-1, and r is a factor of r, but z

A polynomial is monic if coefficient of highest degree term is 1. For example :…

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A polynomial f(z) has the factor-square property (or FSP) if f(z) is a factor of f(r). For instance, g(z) = 1-1 and h(z) =1 have FSP, but k(z) = 1+2 does not. Reascn: z-1 is a factor of r -1, and r is a factor of , but z+2 is not a factor of r +2. Multiplying by a nonzero constant "preserves" FSP, so we restrict attention to poly- nomials that are monic (ie., have 1 as highest-degree coefficient). What patterns do monie FSP polynomials satisfy? To make progress on this topie, investigate the following questions and justify your answers. (a) Are z and z-1 the only monic FSP polynomials of degree 1? (b) List all the monic FSP polynomials of degree 2. To start, note that r, 7-1, -1, and z+z+1 are on that list. Some of them are products of FSP polynomials of smaller degree. For instance, * and -r arise from degree 1 cases. However, r-1 and r +z+1 are new, not expressible as a product of two smaller FSP polynomials. Which terms in your list of degree 2 examples are new? (c) List all the new monic FSP polynomials of degree 3. Note: Some memie FSP polynemiala of dege 3 have complex coelicinta that ae not real. Can you make a similar list in degree 4?

A polynomial f(z) has the factor-square property (or FSP) if f(z) is a factor of f(r). For instance, g(z) = 1-1 and h(z) =1 have FSP, but k(z) = 1+2 does not. Reascn: z-1 is a factor of r -1, and r is a factor of , but z+2 is not a factor of r +2. Multiplying by a nonzero constant "preserves" FSP, so we restrict attention to poly- nomials that are monic (ie., have 1 as highest-degree coefficient). What patterns do monie FSP polynomials satisfy? To make progress on this topie, investigate the following questions and justify your answers. (a) Are z and z-1 the only monic FSP polynomials of degree 1? (b) List all the monic FSP polynomials of degree 2. To start, note that r, 7-1, -1, and z+z+1 are on that list. Some of them are products of FSP polynomials of smaller degree. For instance, * and -r arise from degree 1 cases. However, r-1 and r +z+1 are new, not expressible as a product of two smaller FSP polynomials. Which terms in your list of degree 2 examples are new? (c) List all the new monic FSP polynomials of degree 3. Note: Some memie FSP polynemiala of dege 3 have complex coelicinta that ae not real. Can you make a similar list in degree 4?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.

Topic Video

Question

Answer (a), (b), and (c) only.

A polynomial f(z) has the factor-square property (or FSP) if f(x) is a factor of f(r²).
For instance, g(x) =1-1 and h(z) = r have FSP, but k(z) = 1+2 does not.
Reason: z-1 is a factor of r-1, and r is a factor of r, but z+2 is not a factor of r +2.
Multiplying by a nonzero constant “preserves" FSP, so we restrict attention to poly-
nomials that are monic (ie., have 1 as highest-degree coefficient).
What patterns do monic FSP polynomials satisfy?
To make progress on this topic, investigate the following questions and justify your
answers.
(a) Are z and r - 1 the only monic FSP polynomials of degree 1?
(b) List all the monic FSP polynomials of degree 2.
To start, note that r, r -1, r – 1, and r +1+1 are on that list.
Some of them are products of FSP polynomials of smaller degree. For instance,
7 and r-r arise from degree 1 cases. However, r -1 and r +r+1 are new,
not expressible as a product of two smaller FSP polynomials.
Which terms in your list of degree 2 examples are new?
(c) List all the new monic FSP polynomials of degree 3.
Note: Some monic FSP polynomials of degree 3 have complex coefficienta that are not real.
Can you make a similar list in degree 4?
(d) Are there monic FSP polynomials (of some degree) that have real number
coefficients, but some of those coefficients are not integers?
Explain your reasoning.

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