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Pascal's Triangle

Triangular array of numbers describing coefficients for a binomial raised to a power.

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Polynomial Expansion and Pascal's Triangle

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Pascal's triangle is a number pattern presented in the form of a triangle.  The primary purpose of it is to help calculate the expansion of binomials.  The following figures demonstrate how:

(x+y)^0 &= 1\\(x+y)^1 &= x+y\\(x+y)^2 &= x^2+2y+y^2\\(x+y)^3 &= x^3+3x^2y+3xy^2+y^3

In the above figure, notice how the coefficients of the terms of x match the levels on Pascal's triangle.  We use this pattern to expand binomials of the form (x+y)n.   The nth row of the triangle will give you the number by which each term must be multiplied by.  The term can be found by numbering the terms of the expanded polynomial.  The first term will have the first value (x in this case) to the power of n and the second value (y) to the power of zero.  The second term will have the first value to the power of n-1 and the second value to the power of 1.  This pattern continues until the last term, which will have the first value to the power of zero and the second value to the power of n.

For more on the uses of Pascal's triangle, click here.

Expand the following binomials:

1. (x+2y)5

2. (x2+y3)4

Answers below

1. x5+10x4y+40x3y2+80x2y3+80xy4+32y5

2. x8+4x6y3+6x4y6+4x2y9+y12

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