1963 AHSME Problems/Problem 21

Problem

The expression $x^2-y^2-z^2+2yz+x+y-z$ has:

$\textbf{(A)}\ \text{no linear factor with integer coefficients and integer exponents} \qquad \\ \textbf{(B)}\ \text{the factor }-x+y+z \qquad \\ \textbf{(C)}\ \text{the factor }x-y-z+1 \qquad \\ \textbf{(D)}\ \text{the factor }x+y-z+1 \qquad \\ \textbf{(E)}\ \text{the factor }x-y+z+1$

Solution

Factor the perfect square trinomial. \[x^2 - (y-z)^2 + x + y - z\] Factor the difference of squares. \[(x+y-z)(x-y+z) + x + y - z\] Factor by grouping. \[(x+y-z)(x-y+z+1)\] The answer is $\boxed{\textbf{(E)}}$.

See Also

1963 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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