Question

completely factor the trinomial, if possible.

\\2x^2 + 22x + 48\\

Explanation:

Factor out the greatest common factor

Using the Factoring Polynomials by GCF knowledge point, we first look for the greatest common factor (GCF) of all three terms in the trinomial \(2x^2 + 22x + 48\).
The coefficients are \(2\), \(22\), and \(48\). The GCF of these numbers is \(2\).
We factor \(2\) out of each term:

$$ 2x^2 + 22x + 48 = 2(x^2 + 11x + 24) $$

Find factors for the quadratic trinomial

Using the Factoring Trinomials knowledge point, we now factor the quadratic expression inside the parentheses, \(x^2 + 11x + 24\).
We need to find two integers whose product is \(24\) and whose sum is \(11\).
Let's list the factor pairs of \(24\) and find their sums:

• \(1 \times 24 = 24\) (Sum: \(1 + 24 = 25\))
• \(2 \times 12 = 24\) (Sum: \(2 + 12 = 14\))
• \(3 \times 8 = 24\) (Sum: \(3 + 8 = 11\))

The correct integers are \(3\) and \(8\).

Write the fully factored expression

We write the trinomial \(x^2 + 11x + 24\) as the product of two binomials:

$$ x^2 + 11x + 24 = (x + 3)(x + 8) $$

Including the GCF of \(2\) factored out in the first step, the complete factored form is:

$$ 2(x + 3)(x + 8) $$

Answer:

\(2(x + 3)(x + 8)\)