Using Pascal’s Triangle to Expand Binomials
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
Expand each binomial:
(a + b) 5
1( from")a 5b 0 + 5( from")a 4 b1 + 10( from")a 3b 2 + 10a 2b 3 + 5a1b 4 + 1a 0b 5
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a 5 + 5a 4 b + 10a 3b 2 + 10a 2b 3 + 5ab 4 + b 5
Hints:
• The first term (a) has exponents that are decreasing by one each time and the
second term (b) has exponents that are increasing by one each term.
• The exponents will always have a sum equal to the power of the original
binomial.
• The number of terms in the expansion will be the power of the expression plus
one.
(x "11) 4
1x 4 (("110 )) + 4 x 3 (("11)1 ) + 6x 2 (("11) 2 ) + 4 x1 (("11) 3 ) + 1x 0 (("11) 4 )
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1x 4 + 4 x 3 ("11) + 6x 2 (121) + 4 x1 ("1331) + 1(14641)
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x 4 " 44 x 3 + 726x 2 " 5324 x + 14641
!
!
(x " y) 7
1x 7 (("y 0 )) + 7x 6 (("y)1 ) + 21x 5 (("y) 2 ) + 35x 4 (("y) 3 ) + 35x 3 (("y) 4 ) + 21x 2 (("y) 5 ) + 7x1 (("y) 6 ) + 1x 0 (("y) 7
!
x 7 + 7x 6 ("y) + 21x 5 (y 2 ) + 35x 4 ("y 3 ) + 35x 3 (y 4 ) + 21x 2 ("y 5 ) + 7x1 (y) 6 + ("y 7 )
!
x 7 " 7x 6 y + 21x 5 y 2 " 35x 4 y 3 + 35x 3 y 4 " 21x 2 y 5 + 7xy 6 "y 7
!
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(m + 1) 3
1m 3 (1) 0 + 3m 2 (1)1 + 3m1 (1) 2 + 1m 0 (1) 3
m 3 + 3m 2 + 3m + 1
!