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Chapter 2: Problem 53
Expand the expression. $$ (3-\sqrt{2 x})^{2} $$
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Find all real numbers \(x\) such that $$ x^{6}-8 x^{3}+15=0 $$.
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s(x))^{2} t(x) $$
$$ \text { Suppose } p(x)=2 x^{6}+3 x^{5}+5 $$ (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 \(p\).
A bicycle company finds that its average cost per bicycle for producing \(n\) thousand bicycles is \(a(n)\) dollars, where $$ a(n)=800 \frac{3 n^{2}+n+40}{16 n^{2}+2 n+45} $$ What will be the approximate cost per bicycle when the company is producing many bicycles?
Find all real numbers \(x\) such that $$ x^{6}-3 x^{3}-10=0 $$.
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