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).

See Also

1961 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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All AHSME Problems and Solutions


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