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Chapter 2: Problem 41
Find the maximum value of \(7-2 x-x^{2}\).
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Suppose \(p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n},\) where \(a_{0}, a_{1}, \ldots, a_{n}\) are integers. Suppose \(M\) and \(N\) are nonzero integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\). Show that \(a_{0} / M\) and \(a_{n} / N\) are integers. [Thus to find rational zeros of a polynomial with integer coefficients, we need only look at fractions whose numerator is a divisor of the constant term and whose denominator is a divisor of the coefficient of highest degree. This result is called the Rational Zeros Theorem or the Rational Roots Theorem.]
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s \circ r)(x) $$
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (4 r+5 s)(x) $$
Find a polynomial \(p\) of degree 3 such that \(-1,2,\) and 3 are zeros of \(p\) and \(p(0)=1\).
Write the domain of the given function \(r\) as a union of intervals. $$ r(x)=\frac{5 x^{3}-12 x^{2}+13}{x^{2}-7} $$
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