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Chapter 2: Problem 8
The numbers are too large to be handled by a calculator. These exercises require an understanding of the concepts. Write \(27^{4000}\) as a power of 3 .
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Verify that \(x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)\).
Suppose \(q(x)=2 x^{3}-3 x+1\) (a) Show that the point (2,11) is on the graph of \(q\). (b) Show that the slope of a line containing (2,11) and a point on the graph of \(q\) very close to (2,11) is approximately 21 . [Hint: Use the result of Exercise \(17 .]\)
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 \((0, \infty)\).
$$ \text { Suppose } p(x)=2 x^{6}+3 x^{5}+5 $$ (a) Show that if \(\frac{M}{N}\) is a zero of \(p\), then $$ 2 M^{6}+3 M^{5} N+5 N^{6}=0 $$ (b) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(5 / M\) and \(2 / N\) are integers. (c) Show that the only possible rational zeros of \(p\) $$ \text { are }-5,-1,-\frac{1}{2}, \text { and }-\frac{5}{2} \text { . } $$ (d) Show that no rational number is a zero of \(p\).
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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