Question

the trinomial $x^2 + bx - c$ has factors of $(x + m)(x - n)$, where $m$, $n$, and $b$ are positive. what is the relationship between the values of $m$ and $n$? explain.

Explanation:

Step1: Expand the factored form

$$(x+m)(x-n) = x^2 -nx + mx - mn = x^2 + (m-n)x - mn$$

Step2: Equate coefficients with trinomial

Compare $x^2 + (m-n)x - mn$ to $x^2 + bx - c$. For the $x$-terms:
$$m - n = b$$

Step3: Use positivity of $b$

Since $b>0$, substitute into the equation:
$$m - n > 0$$

Answer:

$m > n$. When expanding the factored form, the coefficient of $x$ is $m-n$, which equals the positive value $b$. This means $m-n$ must be positive, so $m$ is greater than $n$.