Question 253679


{{{(x+1)^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+1)^6}}}, simply follow this procedure:

Write the first coefficient. Multiply that coefficient with the first binomial term {{{x}}} and then the second binomial term {{{1}}}. Repeat this until all of the coefficients have been written.


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



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




However, we're not done yet.



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


{{{1(x)^6(1)^0+(x)^5(1)^1+15(x)(1)+20(x)(1)+15(x)(1)+6(x)(1)+1(x)(1)}}} Looking at the  second term {{{6(x)(1)}}} raise  {{{x}}} to the 5th power and raise {{{1}}} 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 {{{1}}}  are stepping up.



So the fully expanded expression should now look like this:



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



{{{1(x^6)(1)+6(x^5)(1)+15(x^4)(1)+20(x^3)(1)+15(x^2)(1)+6(x^1)(1)+1(x^0)(1)}}} Distribute the exponents



{{{1(x^6)+6(x^5)+15(x^4)+20(x^3)+15(x^2)+6(x)+1}}} Multiply



{{{x^6+6x^5+15x^4+20x^3+15x^2+6x+1}}} Multiply the terms with their coefficients



So {{{(x+1)^6}}} expands and simplifies to {{{x^6+6x^5+15x^4+20x^3+15x^2+6x+1}}}.



In other words, {{{(x+1)^6=x^6+6x^5+15x^4+20x^3+15x^2+6x+1}}}