How do you expand ${(4 x + y)}^{4}$ $(4x+y)^4$ using Pascal’s Triangle?

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Answer 1 · 2015-08-04T18:48:28

Use a combination of a row of Pascal's triangle and a list of descending powers of $4$ $4$ to find:

${(4 x + y)}^{4} = 256 x^{4} + 256 x^{3} y + 96 x^{2} y^{2} + 16 x y^{3} + y^{4}$ $(4x+y)^4 = 256x^4+256x^3y+96x^2y^2+16xy^3+y^4$

Write down the 5th row for Pascal's triangle as a sequence:

$1$ $1$ , $4$ $4$ , $6$ $6$ , $4$ $4$ , $1$ $1$

Write down the powers of $4$ $4$ from $4^{4}$ $4^4$ down to $4^{0}$ $4^0$ as a sequence:

$256$ $256$ , $64$ $64$ , $16$ $16$ , $4$ $4$ , $1$ $1$

Multiply the two sequences together to get:

$256$ $256$ , $256$ $256$ , $96$ $96$ , $16$ $16$ , $1$ $1$

These are the coefficients of the expanded polynomial:

${(4 x + y)}^{4} = 256 x^{4} + 256 x^{3} y + 96 x^{2} y^{2} + 16 x y^{3} + y^{4}$ $(4x+y)^4 = 256x^4+256x^3y+96x^2y^2+16xy^3+y^4$

How do you expand (4x+y)^4(4x+y)4(4x+y)^4 using Pascal’s Triangle?