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Chapter 2: Problem 20
Find a function \(m\) such that \(m(f)\) is the number of miles in \(f\) feet.
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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 \(p\).
Find a number \(b\) such that 3 is a zero of the polynomial \(p\) defined by $$ p(x)=1-4 x+b x^{2}+2 x^{3} $$.
Give an example of polynomials \(p\) and \(q\) such that \(\operatorname{deg}(p q)=8\) and \(\operatorname{deg}(p+q)=2\).
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 each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{2}}{4 x+3} $$
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