How do you use the pascals triangle to expand $(x + 2)^5$ ?

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Answer 1 · 2016-05-30T16:14:48

$(x+2)^5 = x^5+10x^4+40x^3+80x^2+80x+32$

In general:

$(a+b)^5 = ((5),(0))a^5+((5),(1))a^4b+((5),(2))a^3b^2+((5),(3))a^2b^3+((5),(4))ab^4+((5),(5))b^5$

where $((5),(k)) = (5!)/((5-k)!k!)$

These binomial coefficients are found as a row of Pascal's triangle:

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Write out the row of Pascal's triangle that begins $1, 5$ ...

$1, 5, 10, 10, 5, 1$

Write out powers of $2$ up to $2^5$ :

$1, 2, 4, 8, 16, 32$

Multiply the two sequences together:

$1, 10, 40, 80, 80, 32$

These are the coefficients we need:

$(x+2)^5 = x^5+10x^4+40x^3+80x^2+80x+32$

How do you use the pascals triangle to expand (x + 2)^5(x + 2)^5?