How do you use the binomial theorem to expand and simplify the expression #3(x+1)^5-4(x+1)^3#?

1 Answer
Mar 23, 2017

Answer:

The result is #x^5+5x^4+26x^3+18x^2+3x-1#

Details of the expansions follow...

Explanation:

The first binomial will expand into

#ax^5+bx^4+cx^3+dx^2+fx+g#

The trick is coming up with #a, b, c, d, f and g# in as painless a way as possible. You can use Pascal's triangle, or combinations #""_nC_r#, where #n# is 5, and #r# is 0, 1, 2, 3, 4, 5# in that order.

image.slidesharecdn.com/pascalstriangle-140219222731-phpapp02/95/pascals-triangle-1-638.jpg?cb=1392871296

Either way, the coefficients turn out to be 1, 5, 10, 10, 5, 1 in that order.

#x^5+5x^4+10x^3+10dx^2+5x+1#

Now multiply the whole thing by 3.

#3x^5+15x^4+30x^3+30x^2+15x+3#

Subtract from this the second expanded binomial (also get the coefficients in one of the ways mentioned above):

#4(x^3+3x^2+3x+1)#

#4x^3+12x^2+12x+4#

The finished result is found by combining like terms (those in #x^3, x^2, x, etc.)

#x^5+5x^4+26x^3+18x^2+3x-1#

Hope that all makes sense!