Question

factor completely. \\( z^2 - 14z + 24 \\)

Explanation:

Step1: Check for common factors

First, we check if there is a common factor for the coefficients of \(z^2\), \(z\), and the constant term. The coefficients are 1 (for \(z^2\)), -14 (for \(z\)), and 24 (constant). The greatest common divisor of 1, 14, and 24 is 1, so we can't factor out a common number.

Step2: Factor the quadratic

We need to find two numbers that multiply to \(1\times24 = 24\) and add up to -14. Let's list the factor pairs of 24: (1,24), (2,12), (3,8), (4,6). Among these, -2 and -12 multiply to 24 (\((-2)\times(-12)=24\)) and add up to -14 (\((-2)+(-12)=-14\)).
So we can rewrite the middle term using these two numbers:
\(z^2 - 14z + 24=z^2 - 2z - 12z + 24\)

Step3: Group and factor

Group the first two terms and the last two terms:
\((z^2 - 2z)+(- 12z + 24)\)
Factor out the common factors from each group:
\(z(z - 2)-12(z - 2)\)
Now we can factor out \((z - 2)\) from both terms:
\((z - 2)(z - 12)\)

Answer:

\((z - 2)(z - 12)\)