Question 222493



{{{(x-2y)^3}}} 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>




Looking at the row that starts with 1,3, etc, we can see that this row has the numbers:


1, 3, 3, and 1


These numbers will be the coefficients of our expansion. So to expand {{{(x-2y)^3}}}, simply follow this procedure:

Write the first coefficient. Multiply that coefficient with the first binomial term {{{x}}} and then the second binomial term {{{-2y}}}. Repeat this until all of the coefficients have been written.


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



{{{highlight(1)(x)(-2y)+highlight(3)(x)(-2y)+highlight(3)(x)(-2y)+highlight(1)(x)(-2y)}}} Notice how the coefficients are in front of each term.




However, we're not done yet.



{{{1(x)^3(-2y)^0+(x)(-2y)+3(x)(-2y)+3(x)(-2y)+1(x)(-2y)}}} Looking at the first term {{{1(x)(-2y)}}}, raise  {{{x}}} to the 3rd power and raise {{{-2y}}} to the 0th power.


{{{1(x)^3(-2y)^0+(x)^2(-2y)^1+3(x)(-2y)+3(x)(-2y)+1(x)(-2y)}}} Looking at the  second term {{{3(x)(-2y)}}} raise  {{{x}}} to the 2nd power and raise {{{-2y}}} 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 {{{-2y}}}  are stepping up.



So the fully expanded expression should now look like this:



{{{1(x)^3(-2y)^0+3(x)^2(-2y)^1+3(x)^1(-2y)^2+1(x)^0(-2y)^3}}}



{{{1(x^3)(y^0)+3(x^2)(-2y^1)+3(x^1)(4y^2)+1(x^0)(-8y^3)}}} Distribute the exponents



{{{1(x^3)+3(-2x^2y)+3(4xy^2)+1(-8y^3)}}} Multiply



{{{x^3-6x^2y+12xy^2-8y^3}}} Multiply the terms with their coefficients



So {{{(x-2y)^3}}} expands and simplifies to {{{x^3-6x^2y+12xy^2-8y^3}}}.



In other words, {{{(x-2y)^3=x^3-6x^2y+12xy^2-8y^3}}}