Question

factor.
$3z^2 - 2z - 21$

Explanation:

Step1: Find two numbers for ac

For \(3z^2 - 2z - 21\), \(a = 3\), \(c=-21\), so \(ac = 3\times(-21)=-63\). We need two numbers that multiply to \(-63\) and add to \(-2\). These numbers are \(-9\) and \(7\) (since \(-9\times7 = -63\) and \(-9 + 7=-2\)).

Step2: Rewrite the middle term

Rewrite \(-2z\) as \(-9z + 7z\) in the expression:
\(3z^2 - 9z + 7z - 21\)

Step3: Group and factor

Group the first two terms and the last two terms:
\((3z^2 - 9z)+(7z - 21)\)
Factor out the GCF from each group:
\(3z(z - 3)+7(z - 3)\)

Step4: Factor out the common binomial

Now, factor out \((z - 3)\) from both terms:
\((3z + 7)(z - 3)\)

Answer:

\((3z + 7)(z - 3)\)