Question

completely factor the given polynomial, if possible. if the polynomial cannot be factored, indicate
ot factorable\.

\\2x^2 - 14x + 24\\

Explanation:

Factor out the greatest common factor

We begin by identifying the greatest common factor (GCF) of the terms in the polynomial \(2x^2 - 14x + 24\).

Using the Factoring Polynomials by GCF knowledge point:
The coefficients are \(2\), \(-14\), and \(24\). The greatest common divisor of these three numbers is \(2\). We factor \(2\) out of the entire expression:

$$ 2x^2 - 14x + 24 = 2(x^2 - 7x + 12) $$

Factor the remaining trinomial

Next, we factor the quadratic trinomial inside the parentheses, \(x^2 - 7x + 12\).

Using the Factoring Trinomials knowledge point:
We look for two integers whose product is the constant term \(12\) and whose sum is the coefficient of the linear term \(-7\).

• The factors of \(12\) that add up to \(-7\) are \(-3\) and \(-4\), since:

$$ (-3) \cdot (-4) = 12 $$

$$ (-3) + (-4) = -7 $$

Thus, we can write the trinomial as:

$$ x^2 - 7x + 12 = (x - 3)(x - 4) $$

Write the completely factored expression

We combine the factored parts to write the final completely factored polynomial.

We multiply the GCF we factored out in the first step by the factored trinomial:

$$ 2(x - 3)(x - 4) $$

This expression cannot be factored any further.

Answer:

\(2(x - 3)(x - 4)\)