How do I use Pascal's triangle to expand $(3a + b)^4$ ?

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Answer 1 · 2015-07-03T22:24:27

Combine a row of Pascal's triangle with a list of powers of $3$ to find:

$(3a+b)^4 = 81a^4+108a^3b+54a^2b^2+12ab^3+b^4$

Write out the 5th row of Pascal's triangle as a sequence:

$1, 4, 6, 4, 1$

Write out powers of $3$ from $3^4$ down to $3^0$ as a sequence:

$81, 27, 9, 3, 1$

Multiply the two sequences together to get the sequence:

$81, 108, 54, 12, 1$

These are the coefficients of the expansion:

$(3a+b)^4 = 81a^4+108a^3b+54a^2b^2+12ab^3+b^4$

How do I use Pascal's triangle to expand (3a + b)^4(3a + b)^4?