1963 AHSME Problems/Problem 21

Revision as of 14:43, 16 August 2021 by Chessauda (talk | contribs) (Solution 1)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png