August 19, 2007
This session is based on the material in section 2.3 of Kelley's "The Humongous Book of Calculus Problems"
The section presents a series of problems involving factoring polynomial expressions
Problem 2.20

In[33]:=

Factor[18x^2y^5 - 9xy^3]

Out[33]=

-9 (xy^3 - 2 x^2 y^5)

In[34]:=

Simplify[18x^2y^5 - 9xy^3]

Out[34]=

-9 (xy^3 - 2 x^2 y^5)

In[39]:=

FullSimplify[18x^2y^5 - 9xy^3]

Out[39]=

-9 (xy^3 - 2 x^2 y^5)

In[40]:=

FactorTerms[18x^2y^5 - 9xy^3]

Out[40]=

9 (-xy^3 + 2 x^2 y^5)

In[41]:=

FactorTerms[18x^2y^5 - 9xy^3, x]

Out[41]=

9 (-xy^3 + 2 x^2 y^5)

In[42]:=

Collect[18x^2y^5 - 9xy^3, x]

Out[42]=

-9 xy^3 + 18 x^2 y^5

In[43]:=

Collect[18x^2y^5 - 9xy^3, x, y]

Out[43]=

y[-9 xy^3] + x^2 y[18 y^5]

In[44]:=

Decompose[18x^2y^5 - 9xy^3]

Out[44]=

{-9 xy + 18 x^2 y^5, xy^3}

This last expression is the closest to that expected by Kelley.
2.21

In[45]:=

Decompose[21x^5y^9z^6 - 15x^4y^2z^11 + 36x^8y^3z]

Out[45]=

{36 x^8 y^3 z + 21 x^5 y^9 z^6 - 15 x^4 y^2 z^11}

In[46]:=

Factor[21x^5y^9z^6 - 15x^4y^2z^11 + 36x^8y^3z]

Out[46]=

3 x^4 y^2 z (12 x^4 y + 7 x y^7 z^5 - 5 z^10)

Yes. But why did this pull out the common factors but not earlier with #2.20?
Let's try that again.

In[47]:=

Factor[-9xy^3 + 18x^2y^5]

Out[47]=

-9 (xy^3 - 2 x^2 y^5)

Strange.

In[49]:=

Factor[18x^2y^5 - 9xy^3 + 3xy]

Out[49]=

-3 (-xy + 3 xy^3 - 6 x^2 y^5)

In[50]:=

Factor[18 (x^2) (y^5) - 9 (x) (y^3) + 3xy]

Out[50]=

3 (xy - 3 x y^3 + 6 x^2 y^5)

I still don't see why Mathematica doesn't factor out the xy.
Let's move on.

In[51]:=

Factor[x^2 + 13x + 40]

Out[51]=

(5 + x) (8 + x)

Good.

In[52]:=

Factor[x^2 - 7x - 18]

Out[52]=

(-9 + x) (2 + x)

In[53]:=

Factor[x^2 - 49]

Out[53]=

(-7 + x) (7 + x)

In[54]:=

Factor[8a^2 + 125b^2]

Out[54]=

8 a^2 + 125 b^2

In[55]:=

Expand[8a^2 + 125b^2]

Out[55]=

8 a^2 + 125 b^2

In[56]:=

FactorTerms[8a^2 + 125b^2]

Out[56]=

8 a^2 + 125 b^2

In[57]:=

Simplify[8a^2 + 125b^2]

Out[57]=

8 a^2 + 125 b^2

In[58]:=

FullSimplify[8a^2 + 125b^2]

Out[58]=

8 a^2 + 125 b^2

Nope. Mathematica is not able to find the factors of this simple sum of two perfect cubes.

In[59]:=

Factor[4x^3 - 20x^2 - 3x + 15]

Out[59]=

(-5 + x) (-3 + 4 x^2)

Yes. Mathematica had no difficulty with this expression.

In[60]:=

Factor[6x^2 + 7x - 24]

Out[60]=

(-3 + 2 x) (8 + 3 x)

Good. In general Mathematica seems to be able to handle most complex expressions, but there are a few that it misses. Weird.


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