Answer (d) only. A polynomial f(æ) has the factor-square property (or FSP) if f(x) is a factor of f(x?).For instance, g(x) = x – 1 and h(x) = x have FSP, but k(x) = x + 2 does not.Reason: r -1 is a factor of a2 – 1, and r is a factor of x2, but r +2 is not a factor of r2 +2.Multiplying by a nonzero constant “preserves" FSP, so we restrict attention to poly-nomials that are monic (i.e., 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 r and x – 1 the only monic FSP polynomials of degree 1?(b) List all the monic FSP polynomials of degree 2.To start, note that x?, x? – 1, x? – x, and x? + x + 1 are on that list.Some of them are products of FSP polynomials of smaller degree. For instance,x2 and x2 – x arise from degree 1 cases. However, x² – 1 and x2 + x+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 coefficients 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.

Name: Answer (d) only. A polynomial f(æ) has the factor-square property (or
Uploaded: 2024-03-20T06:47:08.000Z
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Description: Answer (d) only. A polynomial f(æ) has the factor-square property (or FSP) if f(x) is a factor of f(x?).For instance, g(x) = x – 1 and h(x) = x have FSP, but k(x) = x + 2 does not.Reason: r -1 is a factor of a2 – 1, and r is a factor of x2, but r +2