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Chapter 2: Problem 94
Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=32 x^{5} $$
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Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{2}}{4 x+3} $$
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r+s)(x) $$
Suppose \(p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n},\) where \(a_{0}, a_{1}, \ldots, a_{n}\) are integers. Suppose \(m\) is a nonzero integer that is a zero of \(p\). Show that \(a_{0} / m\) is an integer. [This result shows that to find integer zeros of a polynomial with integer coefficients, we need only look at divisors of its constant term.]
Suppose \(p\) is a polynomial and \(t\) is a number. Explain why there is a polynomial \(G\) such that $$ \frac{p(x)-p(t)}{x-t}=G(x) $$ for every number \(x \neq t\).
Give an example of polynomials \(p\) and \(q\) such that \(\operatorname{deg}(p q)=8\) and \(\operatorname{deg}(p+q)=2\).
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