Question 282922



{{{(2x+3)^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 {{{(2x+3)^4}}}, simply follow this procedure:

Write the first coefficient. Multiply that coefficient with the first binomial term {{{2x}}} and then the second binomial term {{{3}}}. Repeat this until all of the coefficients have been written.


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



{{{highlight(1)(2x)(3)+highlight(4)(2x)(3)+highlight(6)(2x)(3)+highlight(4)(2x)(3)+highlight(1)(2x)(3)}}} Notice how the coefficients are in front of each term.




However, we're not done yet.



{{{1(2x)^4(3)^0+(2x)(3)+6(2x)(3)+4(2x)(3)+1(2x)(3)}}} Looking at the first term {{{1(2x)(3)}}}, raise  {{{2x}}} to the 4th power and raise {{{3}}} to the 0th power.


{{{1(2x)^4(3)^0+(2x)^3(3)^1+6(2x)(3)+4(2x)(3)+1(2x)(3)}}} Looking at the  second term {{{4(2x)(3)}}} raise  {{{2x}}} to the 3rd power and raise {{{3}}} 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 {{{3}}}  are stepping up.



So the fully expanded expression should now look like this:



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



{{{1(16x^4)(1)+4(8x^3)(3)+6(4x^2)(9)+4(2x^1)(27)+1(x^0)(81)}}} Distribute the exponents



{{{1(16x^4)+4(24x^3)+6(36x^2)+4(54x)+1(81)}}} Multiply



{{{16x^4+96x^3+216x^2+216x+81}}} Multiply the terms with their coefficients



So {{{(2x+3)^4}}} expands and simplifies to {{{16x^4+96x^3+216x^2+216x+81}}}.



In other words, {{{(2x+3)^4=16x^4+96x^3+216x^2+216x+81}}}