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The question is asking for me to expand ${(p+r)}^4$. I know that I have to use Pascal's triangle, the fourth set down, which is $1,4,6$,$4,1.$ My thinking is that I have to use these numbers to solve this through synthetic division, but don't really know what to do from there. Please let me know if I'm completely wrong.

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It's simpler than that. The $1,4,6,4,1$ tell you the coefficents of the $p^4$, $p^3r$, $p^2r^2$, $pr^3$ and $r^4$ terms respectively, so the expansion is just

$$ 1p^4 + 4p^3r + 6p^2r^2 + 4pr^3 + 1r^4 $$

so

$$ p^4 + 4p^3r + 6p^2r^2 + 4pr^3 + r^4 $$

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    $\begingroup$ Bonus exercise for the OP: figure out why this works by starting with the constant polynomial $1$ and repeatedly multiplying it by $(p+r)$. Compare this with the way you calculate the numbers in Pascal's triangle. $\endgroup$ Nov 19, 2015 at 19:50

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