Question 183083



{{{(x-3y)^6}}} Start with the given expression


To expand this, we're going to use binomial expansion. So let's look at Pascal's triangle:
<center>1&nbsp; &nbsp;</center>
<center>1&nbsp; &nbsp;1&nbsp; &nbsp;</center>
<center>1&nbsp; &nbsp;2&nbsp; &nbsp;1&nbsp; &nbsp;</center>
<center>1&nbsp; &nbsp;3&nbsp; &nbsp;3&nbsp; &nbsp;1&nbsp; &nbsp;</center>
<center>1&nbsp; &nbsp;4&nbsp; &nbsp;6&nbsp; &nbsp;4&nbsp; &nbsp;1&nbsp; &nbsp;</center>
<center>1&nbsp; &nbsp;5&nbsp; &nbsp;10&nbsp; &nbsp;10&nbsp; &nbsp;5&nbsp; &nbsp;1&nbsp; &nbsp;</center>
<center>1&nbsp; &nbsp;6&nbsp; &nbsp;15&nbsp; &nbsp;20&nbsp; &nbsp;15&nbsp; &nbsp;6&nbsp; &nbsp;1&nbsp; &nbsp;</center>




Looking at the row that starts with 1,6, etc, we can see that this row has the numbers:


1, 6, 15, 20, 15, 6, and 1


These numbers will be the coefficients of our expansion. So to expand {{{(x-3y)^6}}}, simply follow this procedure:

Write the first coefficient. Multiply that coefficient with the first binomial term {{{x}}} and then the second binomial term {{{-3y}}}. Repeat this until all of the coefficients have been written.



Once that has been done, add up the terms like this:



{{{highlight(1)(x)(-3y)+highlight(6)(x)(-3y)+highlight(15)(x)(-3y)+highlight(20)(x)(-3y)+highlight(15)(x)(-3y)+highlight(6)(x)(-3y)+highlight(1)(x)(-3y)}}} Notice how the coefficients are in front of each term.




However, we're not done yet.



{{{1(x)^6(-3y)^0+(x)(-3y)+6(x)(-3y)+15(x)(-3y)+20(x)(-3y)+15(x)(-3y)+6(x)(-3y)+1(x)(-3y)}}} Looking at the first term {{{1(x)(-3y)}}}, raise  {{{x}}} to the 6th power and raise {{{-3y}}} to the 0th power.



{{{1(x)^6(-3y)^0+(x)^5(-3y)^1+6(x)(-3y)+15(x)(-3y)+20(x)(-3y)+15(x)(-3y)+6(x)(-3y)+1(x)(-3y)}}} Looking at the  second term {{{6(x)(-3y)}}} raise  {{{x}}} to the 5th power and raise {{{-3y}}} to the 1st power.



Continue this until you reach the final term.



Notice how the exponents of {{{x}}} are stepping down and the exponents of {{{-3y}}}  are stepping up.




So the fully expanded expression should now look like this:



{{{1(x)^6(-3y)^0+6(x)^5(-3y)^1+15(x)^4(-3y)^2+20(x)^3(-3y)^3+15(x)^2(-3y)^4+6(x)^1(-3y)^5+1(x)^0(-3y)^6}}}



{{{1(x^6)(y^0)+6(x^5)(-3y^1)+15(x^4)(9y^2)+20(x^3)(-27y^3)+15(x^2)(81y^4)+6(x^1)(-243y^5)+1(x^0)(729y^6)}}} Distribute the exponents



{{{1(x^6)+6(-3x^5y)+15(9x^4y^2)+20(-27x^3y^3)+15(81x^2y^4)+6(-243xy^5)+1(729y^6)}}} Multiply



{{{x^6-18x^5y+135x^4y^2-540x^3y^3+1215x^2y^4-1458xy^5+729y^6}}} Multiply the terms with their coefficients



So {{{(x-3y)^6}}} expands and simplifies to {{{x^6-18x^5y+135x^4y^2-540x^3y^3+1215x^2y^4-1458xy^5+729y^6}}}.



In other words, {{{(x-3y)^6=x^6-18x^5y+135x^4y^2-540x^3y^3+1215x^2y^4-1458xy^5+729y^6}}}