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Chapter 2: Problem 45
Find the equation of the line in the \(x y\) -plane that has slope \(m\) and intersects the \(x\) -axis at \((c, 0)\).
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Give an example of a polynomial \(p\) of degree 6 such that \(p(0)=5\) and \(p(x) \geq 5\) for all real numbers \(\mathcal{X}\).
Suppose \(r\) is the function with domain \((0, \infty)\) defined by $$ r(x)=\frac{1}{x^{4}+2 x^{3}+3 x^{2}} $$ for each positive number \(x\). (a) Find two distinct points on the graph of \(r\). (b) Explain why \(r\) is a decreasing function on \((0, \infty)\). (c) Find two distinct points on the graph of \(r^{-1}\).
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r s)(x) $$
Explain why the polynomial \(p\) defined by $$ p(x)=x^{6}+7 x^{5}-2 x-3 $$ has a zero in the interval (0,1) .
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s \circ r)(x) $$
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