Question 375956



{{{(2x-y)^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 {{{(2x-y)^6}}}, simply follow this procedure:

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


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



{{{highlight(1)(2x)(-y)+highlight(6)(2x)(-y)+highlight(15)(2x)(-y)+highlight(20)(2x)(-y)+highlight(15)(2x)(-y)+highlight(6)(2x)(-y)+highlight(1)(2x)(-y)}}} Notice how the coefficients are in front of each term.




However, we're not done yet.



{{{1(2x)^6(-y)^0+(2x)(-y)+6(2x)(-y)+15(2x)(-y)+20(2x)(-y)+15(2x)(-y)+6(2x)(-y)+1(2x)(-y)}}} Looking at the first term {{{1(2x)(-y)}}}, raise  {{{2x}}} to the 6th power and raise {{{-y}}} to the 0th power.


{{{1(2x)^6(-y)^0+(2x)^5(-y)^1+6(2x)(-y)+15(2x)(-y)+20(2x)(-y)+15(2x)(-y)+6(2x)(-y)+1(2x)(-y)}}} Looking at the  second term {{{6(2x)(-y)}}} raise  {{{2x}}} to the 5th power and raise {{{-y}}} to the 1st power.


Continue this until you reach the final term.



Notice how the exponents of {{{2x}}} are stepping down and the exponents of {{{-y}}}  are stepping up.



So the fully expanded expression should now look like this:



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



{{{1(64x^6)(-y^0)+6(32x^5)(-y)^1+15(16x^4)(-y)^2+20(8x^3)(-y)^3+15(4x^2)(-y)^4+6(2x^1)(-y)^5+1(x^0)(-y)^6}}} Distribute the exponents



{{{1(64x^6)+6(-32x^5y)+15(16x^4y^2)+20(-8x^3y^3)+15(4x^2y^4)+6(-2xy^5)+1(y^6)}}} Multiply



{{{64x^6-192x^5y+240x^4y^2-160x^3y^3+60x^2y^4-12xy^5+y^6}}} Multiply the terms with their coefficients



So {{{(2x-y)^6}}} expands and simplifies to {{{64x^6-192x^5y+240x^4y^2-160x^3y^3+60x^2y^4-12xy^5+y^6}}}.



In other words, {{{(2x-y)^6=64x^6-192x^5y+240x^4y^2-160x^3y^3+60x^2y^4-12xy^5+y^6}}}



If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help">jim_thompson5910@hotmail.com</a>


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Jim