Factor The Trinomial Completely: $x^2 + 13x + 3$.Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. $x^2 + 13x + 3 =$ $\square$ (Type Your Answer In Factored Form.)B. The

**Factor the Trinomial Completely: A Step-by-Step Guide** ===========================================================

What is a Trinomial?

A trinomial is a polynomial expression that consists of three terms. It is a quadratic expression that can be factored into the product of two binomials. In this article, we will focus on factoring the trinomial completely.

Why is Factoring Important?

Factoring is an essential skill in algebra that helps us to simplify complex expressions and solve equations. By factoring a trinomial, we can identify its roots and solve for the variable. This skill is also useful in solving quadratic equations and systems of equations.

How to Factor a Trinomial Completely

To factor a trinomial completely, we need to follow these steps:

Step 1: Identify the Coefficients

The first step is to identify the coefficients of the trinomial. The coefficients are the numbers that multiply the variables. In the trinomial $x^2 + 13x + 3$, the coefficients are 1, 13, and 3.

Step 2: Look for Common Factors

The next step is to look for common factors among the coefficients. In this case, there are no common factors.

Step 3: Use the Factoring Formula

The factoring formula for a trinomial is:

$a(x + b)(x + c)$

where $a$ is the coefficient of the first term, $b$ is the coefficient of the second term, and $c$ is the constant term.

Step 4: Plug in the Values

We can plug in the values of the coefficients into the factoring formula:

$1(x + b)(x + c)$

We need to find the values of $b$ and $c$ that satisfy the equation.

Step 5: Solve for b and c

To solve for $b$ and $c$, we need to find two numbers whose product is equal to the constant term (3) and whose sum is equal to the coefficient of the second term (13). These numbers are 1 and 12.

Step 6: Write the Factored Form

Now that we have found the values of $b$ and $c$, we can write the factored form of the trinomial:

$x^2 + 13x + 3 = (x + 1)(x + 12)$

Example: Factor the Trinomial Completely

Let's factor the trinomial $x^2 + 13x + 3$ completely.

$x^2 + 13x + 3 = (x + 1)(x + 12)$

Q&A

Q: What is a trinomial?

A: A trinomial is a polynomial expression that consists of three terms.

Q: Why is factoring important?

A: Factoring is an essential skill in algebra that helps us to simplify complex expressions and solve equations.

Q: How do I factor a trinomial completely?

A: To factor a trinomial completely, you need to follow these steps: identify the coefficients, look for common factors, use the factoring formula, plug in the values, solve for b and c, and write the factored form.

Q: What is the factoring formula for a trinomial?

A: The factoring formula for a trinomial is $a(x + b)(x + c)$.

Q: How do I find the values of b and c?

A: To find the values of b and c, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the second term.

Q: What is the factored form of the trinomial $x^2 + 13x + 3$?

A: The factored form of the trinomial $x^2 + 13x + 3$ is $(x + 1)(x + 12)$.

Conclusion

Factoring a trinomial completely is an essential skill in algebra that helps us to simplify complex expressions and solve equations. By following the steps outlined in this article, you can factor a trinomial completely and solve for the variable. Remember to identify the coefficients, look for common factors, use the factoring formula, plug in the values, solve for b and c, and write the factored form. With practice, you will become proficient in factoring trinomials completely.