1963 AMC 12 Problem 21

Attempt Problem 21 of the 1963 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1963 AMC 12 solutions, or check the answer key.

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21.

The expression

x^2-y^2-z^2+2yz+x+y-z

has:

no linear factor with integer coefficients and integer exponents

the factor

-x+y+z

the factor

x-y-z+1

the factor

x+y-z+1

the factor

x-y+z+1

Group

y-z

as a single expression

Rewrite the polynomial as

x^2-(y-z)^2+x+(y-z)

Let

u=y-z.

The two groups factor as

\begin{gathered} x^2-u^2=(x+u)(x-u),\\ x+u=(x+u)\cdot1. \end{gathered}

Thus the whole expression is

(x+u)(x-u+1),

and one factor is

x-y+z+1.

Therefore, the correct answer is E.

Problem 21 in other years