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Chapter 2: Problem 11
Find the equation of the line in the \(x y\) -plane with slope 2 that contains the point (7,3) .
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A new snack shop on campus finds that the number of students following it on Twitter at the end of each of its first five weeks in business is 23,89,223 , \(419,\) and \(647 .\) A clever employee discovers that the number of students following the new snack shop on Twitter after \(w\) weeks is \(p(w),\) where \(p\) is defined by $$p(w)=7+3 w+5 w^{2}+9 w^{3}-w^{4}$$ Indeed, with \(p\) defined as above, we have \(p(1)=23,\) \(p(2)=89, p(3)=223, p(4)=419,\) and \(p(5)=647\) Explain why the polynomial \(p\) defined above cannot give accurate predictions for the number of followers on Twitter for weeks far into the future.
Verify that \((x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}\).
Find a polynomial \(p\) of degree 3 such that \(-2,-1,\) and 4 are zeros of \(p\) and \(p(1)=2\).
Give an example of polynomials \(p\) and \(q\) of degree 3 such that \(p(1)=q(1), p(2)=q(2),\) and \(p(3)=q(3),\) but \(p(4) \neq q(4)\).
Suppose \(a, b,\) and \(c\) are integers and that $$ p(x)=a x^{3}+b x^{2}+c x+9 $$ Explain why every zero of \(p\) that is an integer is contained in the set \\{-9,-3,-1,1,3,9\\}.
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