Date post: | 22-Nov-2014 |
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A Tool for Finding thePower of Binomials
What is It?
Basic Definition: •A triangular pattern of numbers in which
each number is equal to the sum of the two numbers immediately above it.
Mathematic Definition:•A geometric arrangement of the binomial
coefficients in a triangle
Write the number 1
Write two more 1’s underneath(forming a triangle)
Write two more 1’s underneath(to the left & right )
Now add the two 1’s and put the sum underneath in the middle
Follow this pattern
1
11
11 2
31 3 1
61 4 4 1...
Each row represents the coefficients of the power of binomials.
(u+v)0 = 1
(u+v)1 = u + v(u+v)2 = u2 + 2uv + v2
(u+v)3 = u3 + 3u2v + 3uv2 + v3
(u+v)4 = u4 + 4u3v + 6u2v2 + 4uv3 + v4
NOTE: We do not write coefficients of 1.
If the coefficients of “1” are included, we can see Pascal’s Triangle forming.
(u+v)0 = 1
(u+v)1 = 1u + 1v
(u+v)2 = 1u2 + 2uv + 1v2
(u+v)3 = 1u3 + 3u2v + 3uv2 + 1v3
(u+v)4 = 1u4 + 4u3v + 6u2v2 + 4uv3 + 1v4
If we change the operation to subtraction, we rotate a “+” & “-” sign in the triangle
(u-v)0 = 1
(u-v)1 = u - v
(u-v)2 = u2 - 2uv + v2
(u-v)3 = u3 - 3u2v + 3uv2 - v3
(u-v)4 = u4 - 4u3v + 6u2v2 - 4uv3 + v4
Examples
Expand the following: (x + 5)3
x3 + 3(x2)(5) + 3(x)(52) + 53
x3 + 15x2 + 75x + 125
Examples
Expand the following: (x - 2)4
x4 – 4(x3)(2) + 6(x2)(22) – 4(x)(23) + 24
x4 – 8x3 + 24x2 – 32x + 16
Examples
Expand the following: (2x + 3)3
(2x)3 + 3(2x)2(3) + 3(2x)(3)2 + 33
8x3 + 36x2 + 54x + 27
Examples
Expand the following: (5x - 7)2
(5x)2 – 2(5x)(7) + 73
25x2 - 70x + 343