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Chapter 2: Problem 54
Expand the expression. $$ (5-\sqrt{3 x})^{2} $$
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Find all real numbers \(x\) such that $$ x^{6}-3 x^{3}-10=0 $$.
Suppose you start driving a car on a hot summer day. As you drive, the air conditioner in the car makes the temperature inside the car \(F(t)\) degrees Fahrenheit at time \(t\) minutes after you started driving, where $$ F(t)=90-\frac{18 t^{2}}{t^{2}+65} $$ (a) What was the temperature in the car when you started driving? (b) What was the approximate temperature in the car 15 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 ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r-s)(x) $$
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (t \circ r)(x) $$
Without doing any calculations or using a calculator, explain why $$ x^{2}+87559743 x-787727821 $$ has no integer zeros. [Hint: If \(x\) is an odd integer, is the expression above even or odd? If \(x\) is an even integer, is the expression above even or odd?]
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