CEMC Fermat, 2024

Algebra Difficulty 2.5 Junior Find the answer Canada

For some integers $m$ and
$n$ , the expression $(x+m)(x+n)$ is equal to a quadratic
expression in $x$ with a constant
term of $-12$ . Which of the following
cannot be a value of $m$ ?

Pick one

Solution

Expanding, $(x+m)(x+n) = x^2 + nx + mx + mn = x^2 + (m+n)x + mn$ .

The constant term of this quadratic expression is $mn$ , and so $mn = -12$ .

Since $m$ and $n$ are integers, they are each divisors of
$-12$ and thus of $12$ .

Of the given possibilities, only $5$ is not a divisor of $12$ , and so $m$ cannot equal $5$ .

We can check that each of the other four choices is a possible value of
$m$ .

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