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The Binomial Theorem

They just get messier -- not much worse.

Expand ( 2x - 3 )^( 5 )
Look at it like ( ( 2x ) + ( -3 ) )^( 5 )

          Our coefficients:

1 , 5 , 10 , 10 , 5 , 1
 

( ( 2x ) + ( -3 ) )^( 5 ) = ( 2x )^( 5 ) + 5( 2x )^( 4 )(-3 )^( 1 ) + 10( 2x )^( 3 )( -3 )^( 2 ) + 10( 2x )^( 2 )( -3 )^( 3 ) + 5( 2x )^( 1 )( -3 )^( 4 ) + ( -3 )^( 5 ) = 32x^( 5 ) - 240x^( 4 ) + 720x^( 3 ) - 1080x^( 2 ) + 810x - 243
 

 

YOUR TURN:

Expand ( a - 2b )^( 4 )

We'll be revisiting these guys in the next chapter.  We just need some more ammunition!