5-7 The Binomial Theorem

Objectives

To expand a binomial using Pascal's Triangle

To use the Binomial Theorem

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There is a connection between the triangular pattern of numbers in the Solve It and the expansion of ( a + b ) n . open eh plus b close to the n . .

Essential Understanding You can use a pattern of coefficients and the pattern

a n , a n − 1 b , a n − 2 b 2 , … , a 2 b n − 2 , a b n − 1 , b n to write the expansion of ( a + b ) n . eh to the n , comma . eh super n minus 1 end super . b comma . eh super n minus 2 end super . b squared , comma dot dot dot comma , eh squared . b super n minus 2 end super . comma eh . b super n minus 1 end super . comma , b to the n . towritetheexpansionof . open , eh plus b , close to the n . .

You can expand ( a + b ) 3 open eh plus b close cubed  using the Distributive Property.

( a + b ) 3 = ( a + b ) ( a + b ) ( a + b ) = a 3 + 3 a 2 b + 3 a b 2 + b 3 open , eh plus b , close cubed . equals . open , eh plus b , close . open , eh plus b , close . open , eh plus b , close . equals , eh cubed , plus 3 , eh squared , b plus 3 eh , b squared , plus , b cubed

To expand the power of a binomial in general, first multiply as needed. Then write the polynomial in standard form.

Consider the expansions of ( a + b ) n open eh plus b close to the n  for the first few values of n:

Row Power Expanded Form Coefficients Only 0 ( a + b ) 0 1 1 1 ( a + b ) 1 1 a 1 + 1 b 1 1 1 2 ( a + b ) 2 1 a 2 + 2 a 1 b 1 + 1 b 2 1 2 1 3 ( a + b ) 3 1 a 3 + 3 a 2 b 1 + 3 a 1 b 2 + 1 b 3 1 3 3 1 4 ( a + b ) 4 1 a 4 + 4 a 3 b 1 + 6 a 2 b 2 + 4 a 1 b 3 + 1 b 4 1 4 6 4 1 table with 6 rows and 4 columns , row1 column 1 , cap row , column 2 cap power , column 3 cap expandedcap form , column 4 cap coefficientscap only , row2 column 1 , 0 , column 2 open , eh plus b , close to the , column 3 1 , column 4 1 , row3 column 1 , 1 , column 2 open , eh plus b , close to the first , column 3 1 , eh to the first , plus 1 , b to the first , column 4 1 1 , row4 column 1 , 2 , column 2 open , eh plus b , close squared , column 3 1 , eh squared , plus 2 , eh to the first , b to the first , plus 1 , b squared , column 4 1 2 1 , row5 column 1 , 3 , column 2 open , eh plus b , close cubed , column 3 1 , eh cubed , plus 3 , eh squared , b to the first , plus 3 , eh to the first , b squared , plus 1 , b cubed , column 4 1 3 3 1 , row6 column 1 , 4 , column 2 open , eh plus b , close to the fourth , column 3 1 , eh to the fourth , plus 4 , eh cubed , b to the first , plus 6 , eh squared , b squared , plus 4 , eh to the first , b cubed , plus 1 , b to the fourth , column 4 1 4 6 4 1 , end table

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