Pascals Triangle A Tool for Finding the Power

Pascal’s Triangle A Tool for Finding the Power of Binomials

PASCAL’S TRIANGLE 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

Discovering Pascal’s Triangle Write the number 1 1 Write two more 1’s underneath (forming a triangle) 1 1 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 2 1 1 3 4 6. . . 1 3 1 4 1

Applying Pascal’s Triangle Each row represents the coefficients of the power of binomials. (u+v)0 = 1 (u+v)1 = u + v (u+v)2 = u 2 + 2 uv + v 2 (u+v)3 = u 3 + 3 u 2 v + 3 uv 2 + v 3 (u+v)4 = u 4 + 4 u 3 v + 6 u 2 v 2 + 4 uv 3 + v 4 NOTE: We do not write coefficients of 1.

Applying Pascal’s Triangle If the coefficients of “ 1” are included, we can see Pascal’s Triangle forming. (u+v)0 = 1 (u+v)1 = 1 u + 1 v (u+v)2 = 1 u 2 + 2 uv + 1 v 2 (u+v)3 = 1 u 3 + 3 u 2 v + 3 uv 2 + 1 v 3 (u+v)4 = 1 u 4 + 4 u 3 v + 6 u 2 v 2 + 4 uv 3 + 1 v 4

Applying Pascal’s Triangle 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 = u 2 - 2 uv + v 2 (u-v)3 = u 3 - 3 u 2 v + 3 uv 2 - v 3 (u-v)4 = u 4 - 4 u 3 v + 6 u 2 v 2 - 4 uv 3 + v 4

PASCAL’S TRIANGLE Examples Expand the following: (x + 5)3 x 3 + 3(x 2)(5) + 3(x)(52) + 53 3 x + 2 15 x + 75 x + 125

PASCAL’S TRIANGLE Examples Expand the following: (x - 2)4 x 4 – 4(x 3)(2) + 6(x 2)(22) – 4(x)(23) + 24 4 x – 3 8 x + 2 24 x – 32 x + 16

PASCAL’S TRIANGLE Examples Expand the following: (2 x + 3)3 (2 x)3 + 3(2 x)2(3) + 3(2 x)(3)2 + 33 3 8 x + 2 36 x + 54 x + 27

PASCAL’S TRIANGLE Examples Expand the following: (5 x - 7)2 (5 x)2 – 2(5 x)(7) + 73 2 25 x - 70 x + 343

Pascal’s Triangle