Question 198002



{{{(x+y)^4}}} 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>




Looking at the row that starts with 1,4, etc, we can see that this row has the numbers:


1, 4, 6, 4, and 1


These numbers will be the coefficients of our expansion. So to expand {{{(x+y)^4}}}, simply follow this procedure:

Write the first coefficient. Multiply that coefficient with the first binomial term {{{x}}} 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)(x)(y)+highlight(4)(x)(y)+highlight(6)(x)(y)+highlight(4)(x)(y)+highlight(1)(x)(y)}}} Notice how the coefficients are in front of each term.




However, we're not done yet.



{{{1(x)^4(y)^0+(x)(y)+6(x)(y)+4(x)(y)+1(x)(y)}}} Looking at the first term {{{1(x)(y)}}}, raise  {{{x}}} to the 4th power and raise {{{y}}} to the 0th power.


{{{1(x)^4(y)^0+(x)^3(y)^1+6(x)(y)+4(x)(y)+1(x)(y)}}} Looking at the  second term {{{4(x)(y)}}} raise  {{{x}}} to the 3rd power and raise {{{y}}} 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 {{{y}}}  are stepping up.



So the fully expanded expression should now look like this:



{{{1(x)^4(y)^0+4(x)^3(y)^1+6(x)^2(y)^2+4(x)^1(y)^3+1(x)^0(y)^4}}}



{{{1(x^4)(y^0)+4(x^3)(y^1)+6(x^2)(y^2)+4(x^1)(y^3)+1(x^0)(y^4)}}} Distribute the exponents



{{{1(x^4)+4(x^3y)+6(x^2y^2)+4(xy^3)+1(y^4)}}} Multiply



{{{x^4+4x^3y+6x^2y^2+4xy^3+y^4}}} Multiply the terms with their coefficients



So {{{(x+y)^4}}} expands and simplifies to {{{x^4+4x^3y+6x^2y^2+4xy^3+y^4}}}.



In other words, {{{(x+y)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4}}}