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

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Answer 1 · 2015-07-03T22:24:27

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

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

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

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

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

$81, 27, 9, 3, 1$ $81, 27, 9, 3, 1$

Multiply the two sequences together to get the sequence:

$81, 108, 54, 12, 1$ $81, 108, 54, 12, 1$

These are the coefficients of the expansion:

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

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