12 of 78 Items .... Course: ALG2

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You are asked to find the roots and factors of the following polynomial function:

\( f(x) = x^4 - 2x^3 - 18x^2 + 6x + 45 \)

By the rational root theorem, possible rational roots are
\( \pm ( 1, 3, 5, 9, 15, 45 ) \).

In order to minimize your effort, you know that you should begin with the possibility that is most likely to be a root. Which roots are most likely?

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Algebra 2:
Write a function for this graph ...

Estimation:
What are some possible values for a,b,c,d?
If I told you it also went through (0,54), what might the leading coefficient be?

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How many different solutions can we find?

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This question is from the NY Regents test, and discussed by Patrick Honner in his long-running series, "Are These Tests Any Good?"

Can your students find what is wrong with it?


How should they fix it?

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So ... I've seen a couple of YouTube videos that feature songs about the Quadratic Formula. I often see it written like this:

\(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}
\)

and it occurs to me that I've always written it this way:

\(x = \dfrac{-b}{2a} \pm \dfrac{\sqrt{b^2-4ac}}{2a}\)

Which one is better?

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\( n^2 - 1 \) is always a multiple of \(n - 1\)

What would this proof look like?

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Population of organism after n minutes is \(P(n) = k*9^{(\frac{n}{2})}\).

What is the growth factor?
... algebraically stated, what is the ratio of P(t+1):P(t)?

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This 1961-era 1MB hard drive is not large enough to store this picture before it was condensed and converted to a JPG. It is the size of a counter-top taoster oven.




This 28MB IBM 1301 machine cost $110,000.00, weighed half a ton, and was the size of three large refrigerators side by side
.



And now you have this to carry around in your pocket.



Go here:
http://www.jcmit.com/flashprice.htm

Can you compose a simple rule about prices over time?
Data copied here:


X (decimal date),Y (US$/MB)
2003.17,0.258
2003.58,0.191
2003.75,0.219
2004,0.191
2004.17,0.238
2004.33,0.266
2004.42,0.223
2005.5,0.0762
2006.33,0.0219
2006.67,0.0156
2006.83,0.0146
2007,0.0122
2007.17,0.0104
2007.33,0.0105
2007.5,0.00976
2007.67,0.00805
2007.75,0.00763
2007.83,0.00732
2007.92,0.00763
2008,0.00458
2008.08,0.00463
2008.33,0.00415
2008.5,0.00354
2008.58,0.00305
2008.67,0.00311
2008.83,0.00183
2008.92,0.00214
2009,0.00162
2009.08,0.00168
2009.25,0.00204
2009.42,0.00214
2009.5,0.00195
2009.58,0.00177
2009.75,0.00214
2009.92,0.00226
2010,0.00244
2010.08,0.00226
2010.17,0.00244
2010.33,0.00201
2010.5,0.00193
2010.58,0.00176
2010.75,0.00177
2010.83,0.00146
2010.92,0.00119
2011,0.001403
2011.08,0.001251
2011.33,0.001342
2011.42,0.001219
2011.67,0.001007
2011.75,0.001037
2012,0.000824
2012.08,0.000824
2012.25,0.000793
2012.33,0.000702
2012.58,0.00058
2012.62,0.00061
2012.83,0.00061
2013,0.000458
2013.08,0.000518
2013.33,0.00058
2013.42,0.000564
2013.58,0.000564
2013.62,0.000564
2013.75,0.000534
2013.83,0.000542
2013.92,0.000548
2014.08,0.000457
2014.17,0.000473
2014.25,0.000366
2014.42,0.000389
2014.58,0.000335
2014.67,0.000345

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Is it possible to find a positive integer value of  n so that this quotient is an integer?

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Can you find values for y that will make each answer true? Generalize the rules in effect here.

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What can we do with this? There doesn't seem to be enough information.


3

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This problem, from Robert Kaplinsky, asks for you to fill in the spaces.




Being the ornery sort, I wondered if there were other coefficients that we could choose for the cubic that might give multiple sets of answers?