A polynomial f(x) has the factor-square property (or FSP) if f(x) is a factor of f(x2)....

 

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A polynomial f(x) has the factor-square property (or FSP) if f(x) is a factor of f(x2). 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 r2- 1, and r is a factor of a2, but r+ 2 is not a factor of a2 +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 your answers. (a) Are r 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 x2, x2 - 1, x2 – x, and a2 + x +1 are on that list. Some of them are products of FSP polynomials of smaller degree. For instance, and a2-x arise from degree 1 cases. However, a2-1 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 monic FSP polynomials of degree 3. Which of those are new? Can you make a similar list in degree 4? (d) Answers to the previous questions may depend on what coefficients are allowed. List the monic FSP polynomials of degree 3 that have integer coefficients. Separately list those (if any) with complex number coefficients that are not all integers. Can you make similar lists for degree 4? Are there examples of monic FSP polynomials with real number coefficients that are not all integers? A polynomial f(x) has the factor-square property (or FSP) if f(x) is a factor of f(x2). 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 r2- 1, and r is a factor of a2, but r+ 2 is not a factor of a2 +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 your answers. (a) Are r 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 x2, x2 - 1, x2 – x, and a2 + x +1 are on that list. Some of them are products of FSP polynomials of smaller degree. For instance, and a2-x arise from degree 1 cases. However, a2-1 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 monic FSP polynomials of degree 3. Which of those are new? Can you make a similar list in degree 4? (d) Answers to the previous questions may depend on what coefficients are allowed. List the monic FSP polynomials of degree 3 that have integer coefficients. Separately list those (if any) with complex number coefficients that are not all integers. Can you make similar lists for degree 4? Are there examples of monic FSP polynomials with real number coefficients that are not all integers?

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