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Chapter 2: Problem 52
Expand the expression. $$ (5+\sqrt{x})^{2} $$
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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 \(r\) 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 \(r^{-1}\).
Find a polynomial \(p\) of degree 3 such that \(-2,-1,\) and 4 are zeros of \(p\) and \(p(1)=2\).
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?
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.]
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