Questions: Factor the trinomial completely. x^2+18x+13

Transcript text: Factor the trinomial completely. \[ x^{2}+18 x+13 \]

Solution

Solution Steps

To factor the trinomial \(x^2 + 18x + 13\), we need to determine if there are two numbers that multiply to 13 (the constant term) and add up to 18 (the coefficient of the linear term). If such numbers exist, the trinomial can be factored into two binomials. If not, the polynomial is prime.

Step 1: Identify the Trinomial

We start with the trinomial \(x^2 + 18x + 13\). Our goal is to factor this expression completely.

Step 2: Check for Factorable Form

To factor the trinomial, we need to find two numbers that multiply to the constant term \(13\) and add up to the linear coefficient \(18\). The pairs of factors of \(13\) are \((1, 13)\) and \((-1, -13)\). However, none of these pairs add up to \(18\).

Step 3: Determine if the Polynomial is Prime

Since there are no pairs of numbers that satisfy both conditions, we conclude that the trinomial cannot be factored into simpler binomials. Therefore, the polynomial is prime.

Final Answer

The polynomial is prime, so the answer is: \\(\boxed{\text{The polynomial is prime.}}\\)

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