1961 AHSME Problems/Problem 5

Problem

Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$ . Then $S$ equals:

$\textbf{(A)}\ (x-2)^4 \qquad \textbf{(B)}\ (x-1)^4 \qquad \textbf{(C)}\ x^4 \qquad \textbf{(D)}\ (x+1)^4 \qquad \textbf{(E)}\ x^4+1$

Solution

Let $y = x-1$ . Substitution results in $S = y^4 + 4y^3 + 6y^2 + 4y + 1$ $S = (y+1)^4$ Substituting back results in $S = x^4$ The answer is $\boxed{\textbf{(C)}}$ . This problem can also be solved by traditionally expanding and combining like terms (though it would take much longer).

1961 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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1961 AHSME Problems/Problem 5

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