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...

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

Extracted text: 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.