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Chapter 2: Problem 47
Expand the expression. $$ (2-3 \sqrt{5})^{2} $$
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$$ \text { Suppose } p(x)=2 x^{6}+3 x^{5}+5 $$ (a) Show that if \(\frac{M}{N}\) is a zero of \(p\), then $$ 2 M^{6}+3 M^{5} N+5 N^{6}=0 $$ (b) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(5 / M\) and \(2 / N\) are integers. (c) Show that the only possible rational zeros of \(p\) $$ \text { are }-5,-1,-\frac{1}{2}, \text { and }-\frac{5}{2} \text { . } $$ (d) Show that no rational number is a zero of \(p\).
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r s)(x) $$
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s(x))^{2} t(x) $$
A textbook states that the rabbit population on a small island is observed to be $$ 1000+120 t-0.4 t^{4} $$ where \(t\) is the time in months since observations of the island began. Explain why the formula above cannot correctly give the number of rabbits on the island for large values of \(t\).
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r t)(x) $$
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