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Chapter 2: Problem 33
Find the slant asymptote of each rational function. $$F(x)=\frac{x^{3}-1}{x^{2}}$$
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INSCRIBED QUADRILATERAL Isaac Newton discovered that if a quadrilateral with sides of lengths \(a, b\) \(c,\) and \(x\) is inscribed in a semicircle with diameter \(x\) then the lengths of the sides are related by the following equation. $$x^{3}-\left(a^{2}+b^{2}+c^{2}\right) x-2 a b c=0$$ Given \(a=6, b=5,\) and \(c=4,\) find \(x .\) Round to the nearest hundredth.
Find a polynomial function \(P(x)\) with real coefficients that has the indicated zeros and satisfies the given conditions. Verify that \(P(x)=x^{3}-x^{2}-i x^{2}-9 x+9+9 i\) has \(1+i\) as a zero and that its conjugate \(1-i\) is not a zero. Explain why this does not contradict the Conjugate Pair Theorem.
When we think of the cube root of \(8, \sqrt[3]{8},\) we normally mean the real cube root of 8 and write \(\sqrt[3]{8}=2 .\) However, there are two other cube roots of 8 that are complex numbers. Verify that \(-1+i \sqrt{3}\) and \(-1-i \sqrt{3}\) are cube roots of 8 by showing that \((-1+i \sqrt{3})^{3}=8\) and \((-1-i \sqrt{3})^{3}=8\).
Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=x^{5}-32$$
Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=x^{4}-1$$
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