How do you expand the binomial #(3x-1)^4#?

1 Answer
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Sep 16, 2016

Using Pascal's triangle gives #81x^4-108x^3+54x^2-12x+1#.

Explanation:

To expand #(3x-1)^4, use row 4 of Pascal's triangle.
The top row is row zero, the next is row 1, the next is row 2, etc.

#color(white)(AAAAAAA)1#
#color(white)(AAAAA)1color(white)(AA)1#
#color(white)(AAA)1color(white)(AA)2color(white)(A)1#
#color(white)(A)1color(white)(AA)3color(white)(AA)3 color(white)(A)1#
#1color(white)(AA)4color(white)(AA)6color(white)(A)4color(white)(AA)1#

Row 4 is 14641. These will be used as the coefficients of each term of the expansion.

To expand #(a-b)^4, use the following process:

#1a^4b^0-4a^3b^1+6a^2b^2-4a^1b^3+1a^0b^4#

Note that the sum of the exponents is 4, because we are finding the 4th power of the binomial. The first term contains #a^4# and #b^0 #, the second #a^3# and #b^1#, the third #a^2# and #b^2#, etc.

Also, the signs alternate because we are expanding #(a-b)#. If we were expanding #(a+b)#, the signs would all be positive.

In our example #a=3x# and #b=1#.

#1(3x)^4(1)^0-4(3x)^3(1)^1+6(3x)^2(1)^2-4(3x)^1(1)^3+1(3x)^0(1)^4#

#81x^4-108x^3+54x^2-12x+1#

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