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