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Chapter 2: Problem 16
Simplify the given expression by writing it as a power of a single variable. $$ y^{4}\left(y^{3}\right)^{5} $$
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Let \(p\) be the polynomial defined by $$p(x)=x^{6}-87 x^{4}-92 x+2$$. (a)Evaluate \(p(-2), p(-1), p(0),\) and \(p(1)\). (b) Explain why the results from part (a) imply that \(p\) has a zero in the interval (-2,-1) and \(p\) has a zero in the interval (0,1) . (c) Show that \(p\) has at least four zeros in the interval [-10,10] . [Hint: We already know from part ( \(b\) ) that \(p\) has at least two zeros is the interval [-10,10] . You can show the existence of other zeros by finding integers \(n\) such that one of the numbers \(p(n)\), \(p(n+1)\) is positive and the other is negative.]
Suppose \(q\) is a polynomial of degree 4 such that $$ \begin{array}{r} q(0)=-1 . \text { Define } p \text { by } \\ \qquad p(x)=x^{5}+q(x) . \end{array} $$ Explain why \(p\) has a zero on the interval \((0, \infty)\).
Suppose $$r(x)=\frac{x+1}{x^{2}+3} \quad \text { and } \quad s(x)=\frac{x+2}{x^{2}+5}$$ Find two distinct numbers \(x\) such that \(s(x)=\frac{1}{8}\).
Suppose you start driving a car on a chilly fall day. As you drive, the heater in the car makes the temperature inside the car \(F(t)\) degrees Fahrenheit at time \(t\) minutes after you started driving, where $$ F(t)=40+\frac{30 t^{3}}{t^{3}+100} $$ (a) What was the temperature in the car when you started driving? (b) the car ten minutes after you started driving? (c) What will be the approximate temperature in the car after you have been driving for a long time?
Write the indicated expression as \(a\) polynomial. $$ (q(x))^{2} s(x) $$
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