Intermediate Algebra (12th Edition)

$(m-7)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $m^2-10m+21 ,$ find two numbers, $m_1$ and $m_2,$ whose product is $c$ and whose sum is $b$ in the quadratic expression $x^2+bx+c.$ Then, express the factored form as $(x+m_1)(x+m_2).$ $\bf{\text{Solution Details:}}$ In the given expression, the value of $c$ is $21$ and the value of $b$ is $-10 .$ The possible pairs of integers whose product is $c$ are \begin{array}{l}\require{cancel} \{ 1,21 \}, \{ 3,7 \}, \{ -1,-21 \}, \{ -3,-7 \} .\end{array} Among these pairs, the one that gives a sum of $b$ is $\{ -3,-7 \}.$ Hence, the factored form of the given expression is \begin{array}{l}\require{cancel} (m-3)(m-7) .\end{array} The missing factor in the given expression is $(m-7) .$