Download EXPANDING BINOMIALS USING PASCAL`S TRIANGLE

EXPANDING BINOMIALS USING PASCAL’S TRIANGLE
n
Expansions of ( x + y ) are:
( x + y )0 = 1
( x + y)1 = x + y
( x + y)2 = x 2 + 2 xy + y 2
( x + y)3 = x3 + 3x 2 y + 3xy 2 + y 3
( x + y)4 = x 4 + 4 x3 y + 6 x 2 y 2 + 4 xy 3 + y 4
( x + y)5 = x5 + 5x 4 y + 10 x3 y 2 + 10 x 2 y 3 + 5 xy 4 + y 5
( x + y)6 = x 6 + 6 x 5 y + 15x 4 y 2 + 20x 3 y 3 + 15x 2 y 4 + 6 xy5 + y 6
( x + y)7 = x 7 + 7 x6 y + 21x5 y 2 + 35x 4 y3 + 35 x3 y 4 + 21x 2 y 5 + 7 xy 6 + y 7
( x + y )8 = x8 + 8x 7 y + 28x 6 y 2 + 56x 5 y 3 + 70x 4 y 4 + 56x 3 y 5 + 28x 2 y 6 + 8 xy 7 + y 8
( x + y)9 = x 9 + 9 x8 y + 36x 7 y 2 + 84x 6 y 3 + 126x 5 y 4 + 126x 4 y 4 + 36x 3 y 5 + 9 x 2 y 6 + y 7
( x + y)10 = x10 + 10 x9 y + 45x8 y 2 + 120 x7 y 3 + 210 x6 y 4 + 252 x5 y 5 + 210 x 4 y 6 + 120 x3 y 7 + 45 x 2 y8 + 10 xy 9 + y10
etc
Patterns that appear in these expansions:
1. Each expansion has one more term than the power of the binomial.
2. The degree of each term in each expansion is equal to the exponent of the binomial that is being expanded.
5
For example in the expansion of ( x + y ) , the sum of the exponents in each term is 5
+ 4=5
3+ 2=5
2+ 3=5
1
4
3 2
2 3
( x + y ) = x + 5 x y + 10 x y + 10 x y + 5 xy 4 + y 5
4 +1= 5
5
5
3. The first term in each expansion is x raised to the power of the binomial, and the last term in each expansion is
y raised to the power of the binomial.
4. The exponent of x decreases by 1 in each successive term. The exponent of
5
term. For example the expansion of ( x + y ) , is
( x + y)5 = x
5
y0 + 5x
4
y1 + 10x 3 y 2 + 10 x
2
y 3 + 5x 1 y 4 + x
0
y5
y increases by 1 in each successive
The coefficients can be written in a triangular array called Pascal’s Triangle, named after the French mathematician
and philosopher Blaise Pascal (1623-1662).
PASCAL’S TRIANGLE
etc
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Each row begins and ends with 1.
The numbers of elements (terms) in a row is 1 more than the row number. For e.g., row 4
has 5 terms, row 5 has 6 terms, row 6 has 7 terms etc.
Each number in the triangle is the sum of the two directly above it.
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The triangle is symmetrical about the vertical line passing through its apex.
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The triangle continues indefinitely.
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Interesting facts
If the 2nd number in a row is a prime number then all the numbers in that row (excluding the 1's) are divisible by it.
For example, in row 7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.
The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the number of the row.
For example:
20 = 1
21 = 2 = 1 + 1
22 = 4 = 1 + 2 + 1
23 = 8 = 1 + 3 = 3 + 1
2 4 = 16 = 1 + 4 + 6 + 4 + 1