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Chapter 0: Problem 35
Factor each trinomial, or state that the trinomial is prime. $$2 x^{2}+3 x y+y^{2}$$
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Factor completely. $$x^{2 n}+6 x^{n}+8$$
What is an algebraic expression? Give an example with your explanation.
In one short sentence, five words or less, explain what $$ \frac{\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}}{\frac{1}{x^{4}}+\frac{1}{x^{5}}+\frac{1}{x^{6}}} $$ does to each number \(x\).
a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?
Factor completely. $$(x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}}$$
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