Question 101585
I have tried this using our book and I just don't udestand the process. My teacher is of no help, she just keeps pointing me to our book. 
Factor each polynomial completely. 
a) x^3y+2x^2y^2+xy^3 
b) -4w^3-16w^2+20w 
c) 3x^2 – 17x + 10 
Thank you so much for any help you can provide me with. :) 
d) -36a^2b + 21ab^2 – 3b^3


a) {{{x^3y+2x^2y^2+xy^3}}} by inspection, we see that we can take an {{{xy}}} out of each term so lets do that:

{{{xy(x^2+2xy+y^2)}}}  Again, by inspection, we see that {{{x^2+2xy+y^2}}} is a perfect square{{{(x+y)^2}}}  but we can also do the following:

{{{xy(x^2+xy+xy+y^2)}}} and we can re-write this as follows:

{{{xy(x(x+y)+y(x+y))}}} now we can factor out an{{{(x+y)}}} and we get:

{{{xy(x+y)(x+y)}}} or
{{{xy(x+y)^2}}}------------------ans


b) {{{-4w^3-16w^2+20w }}}  by inspection, we see that we can take a {{{-4w}}}out of each term so lets do that:

{{{-4w(w^2+4w-5)}}}  Now we observe that {{{w^2+4w-5}}} is a quadratic in standard form.  When we have a quadratic in standard form and the A coefficient is 1 then the B coefficient is the sum of the factors of the C coefficent.  What are the factors of the C coefficient??  They can only be {{{(-5 and +1)}}} or {{{(+5 and -1)}}}.  By inspection we see that +5 and -1=+4.  So now we know that: {{{w^2+4w-5}}}={{{(w+5)(w-1)}}}. Now putting it all back together, we have:

{{{-4w(w-5)(w+1)}}}---------------------ans 

c) {{{3x^2 – 17x + 10}}}  Here are the possibilities:
By inspection, we see that if this quadratic can be factored, then the factors must be of the form (a-b)(c-d) or (c-b)(a-d).  Why???  Because the last term is positive and the middle term is negative.  So here are the possibilities:
1.  {{{(3x-10)(x-1)}}}
2.  {{{(x-10)(3x-1)}}}
3.  {{{(3x-5)(x-2)}}}
4.  {{{(x-5)(3x-2)}}}

Expanding each of the above using the FOIL crutch (First, Inner, Outer, Last) we get:

1.{{{3x^2-10x-3x+10}}}-----------------NO!!! inner term is -13x
2.{{{3x^2-30x-x+10}}}------------------NO!!! inner term is -31x
3.{{{3x^2-5x-2x+10}}}-------------------NO!!!! inner term is -7x
4.{{{3x^2-15x-2x+10}}}----------------BINGO!!!!!!!

{{{(x-5)(3x-2)}}}-----------------------ans

d) {{{-36a^2b+21ab^2-3b^3}}}   By inspection, we see that we can take a -3b out of each term:

{{{-3b(12a^2-7ab+b^2)}}} Now use the approach that was used in (c) above and find the factors for {{{12a^2-7ab+b^2}}} .  I bet you can do it!!!!!


Hope this helps---ptaylor