Question

completely factor the trinomial, if possible.

\\x^2 + 45x + 14\\

Explanation:

Identify the trinomial to factor

We are given the quadratic trinomial expression:

$$ x^2 + 45x + 14 $$

Our goal is to factor this trinomial completely over the integers, if possible.

Determine the factoring strategy

Using the Factoring Trinomials concept, we look for two integers whose product is the constant term and whose sum is the coefficient of the linear term.

• Target product: \(14\)
• Target sum: \(45\)

Find factors with given sum and product

Using the Finding Factors with Given Sum and Product concept, we list all integer factor pairs of \(14\) and calculate their sums:

• \(1 \times 14 = 14\), sum: \(1 + 14 = 15\)
• \((-1) \times (-14) = 14\), sum: \(-1 + (-14) = -15\)
• \(2 \times 7 = 14\), sum: \(2 + 7 = 9\)
• \((-2) \times (-7) = 14\), sum: \(-2 + (-7) = -9\)

Analyze the factor pairs

None of the integer factor pairs of \(14\) add up to the required linear coefficient of \(45\).
Since no such integers exist, the trinomial cannot be factored over the integers.

Conclude the factoring status

A polynomial that cannot be factored into lower-degree polynomials with integer coefficients is called prime.
Therefore, the trinomial is prime (not factorable).

Answer:

Not factorable (prime)