The Factored Form of the Trinomial

Oct 13, 2025 by miodeya.com 127 views

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Introduction

Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will focus on factoring the trinomial $x^2 - 3x - 10$ . Factoring trinomials is an essential skill in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics.

What is a Trinomial?

A trinomial is a polynomial expression that consists of three terms. It can be written in the form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. In the given trinomial $x^2 - 3x - 10$ , the coefficients are $a = 1$ , $b = -3$ , and $c = -10$ .

The Factoring Process

Factoring a trinomial involves finding two binomials whose product equals the original trinomial. The factoring process can be broken down into several steps:

Step 1: Identify the Coefficients

The first step in factoring a trinomial is to identify the coefficients $a$ , $b$ , and $c$ . In the given trinomial $x^2 - 3x - 10$ , the coefficients are $a = 1$ , $b = -3$ , and $c = -10$ .

Step 2: Determine the Signs of the Binomials

The next step is to determine the signs of the binomials. If the product of the two binomials is positive, then both binomials must have the same sign. If the product is negative, then one binomial must have a positive sign, and the other must have a negative sign.

Step 3: Find the Factors of the Constant Term

The constant term in the trinomial is $-10$ . We need to find two numbers whose product equals $-10$ and whose sum equals the coefficient of the linear term, which is $-3$ .

Step 4: Write the Factored Form

Once we have found the factors of the constant term, we can write the factored form of the trinomial.

Factoring the Trinomial $x^2 - 3x - 10$

Let's apply the factoring process to the trinomial $x^2 - 3x - 10$ .

Step 1: Identify the Coefficients

The coefficients of the trinomial are $a = 1$ , $b = -3$ , and $c = -10$ .

Step 2: Determine the Signs of the Binomials

Since the product of the two binomials is negative, one binomial must have a positive sign, and the other must have a negative sign.

Step 3: Find the Factors of the Constant Term

We need to find two numbers whose product equals $-10$ and whose sum equals $-3$ . The factors of $-10$ are $-1$ and $10$ , but their sum is $9$ , not $-3$ . The factors of $-10$ are also $1$ and $-10$ , but their sum is $-9$ , not $-3$ . However, the factors of $-10$ are also $-2$ and $5$ , and their sum is $3$ , not $-3$ . But the factors of $-10$ are also $-5$ and $2$ , and their sum is $-3$ .

Step 4: Write the Factored Form

Now that we have found the factors of the constant term, we can write the factored form of the trinomial.

The Factored Form of the Trinomial

The factored form of the trinomial $x^2 - 3x - 10$ is $(x - 5)(x + 2)$ .

Conclusion

Factoring a trinomial involves expressing a quadratic expression as a product of two binomials. The factoring process involves identifying the coefficients, determining the signs of the binomials, finding the factors of the constant term, and writing the factored form. In this article, we have applied the factoring process to the trinomial $x^2 - 3x - 10$ and found its factored form to be $(x - 5)(x + 2)$ .

Examples and Exercises

Example 1

Factor the trinomial $x^2 + 5x + 6$ .

Solution

The factored form of the trinomial $x^2 + 5x + 6$ is $(x + 2)(x + 3)$ .

Example 2

Factor the trinomial $x^2 - 7x - 18$ .

Solution

The factored form of the trinomial $x^2 - 7x - 18$ is $(x - 9)(x + 2)$ .

Exercise 1

Factor the trinomial $x^2 + 2x - 15$ .

Solution

The factored form of the trinomial $x^2 + 2x - 15$ is $(x + 5)(x - 3)$ .

Exercise 2

Factor the trinomial $x^2 - 4x - 21$ .

Solution

The factored form of the trinomial $x^2 - 4x - 21$ is $(x - 7)(x + 3)$ .

Tips and Tricks

Tip 1

When factoring a trinomial, always start by identifying the coefficients.

Tip 2

When determining the signs of the binomials, remember that if the product of the two binomials is positive, then both binomials must have the same sign.

Tip 3

When finding the factors of the constant term, look for two numbers whose product equals the constant term and whose sum equals the coefficient of the linear term.

Tip 4

When writing the factored form, make sure to include the correct signs of the binomials.

Common Mistakes

Mistake 1

Not identifying the coefficients correctly.

Mistake 2

Not determining the signs of the binomials correctly.

Mistake 3

Not finding the factors of the constant term correctly.

Mistake 4

Not writing the factored form correctly.

Conclusion

Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. The factoring process involves identifying the coefficients, determining the signs of the binomials, finding the factors of the constant term, and writing the factored form. In this article, we have applied the factoring process to the trinomial $x^2 - 3x - 10$ and found its factored form to be $(x - 5)(x + 2)$ .

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Introduction

Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In our previous article, we discussed the factoring process and applied it to the trinomial $x^2 - 3x - 10$ . In this article, we will answer some frequently asked questions about factoring trinomials.

Q&A

Q: What is the first step in factoring a trinomial?

A: The first step in factoring a trinomial is to identify the coefficients $a$ , $b$ , and $c$ . This involves looking at the trinomial and identifying the values of $a$ , $b$ , and $c$ .

Q: How do I determine the signs of the binomials?

A: To determine the signs of the binomials, you need to look at the product of the two binomials. If the product is positive, then both binomials must have the same sign. If the product is negative, then one binomial must have a positive sign, and the other must have a negative sign.

Q: How do I find the factors of the constant term?

A: To find the factors of the constant term, you need to look for two numbers whose product equals the constant term and whose sum equals the coefficient of the linear term.

Q: What is the factored form of the trinomial $x^2 + 5x + 6$ ?

A: The factored form of the trinomial $x^2 + 5x + 6$ is $(x + 2)(x + 3)$ .

Q: What is the factored form of the trinomial $x^2 - 7x - 18$ ?

A: The factored form of the trinomial $x^2 - 7x - 18$ is $(x - 9)(x + 2)$ .

Q: How do I factor a trinomial with a negative constant term?

A: To factor a trinomial with a negative constant term, you need to look for two numbers whose product equals the constant term and whose sum equals the coefficient of the linear term. Then, you can write the factored form using the correct signs of the binomials.

Q: What is the difference between factoring and simplifying a trinomial?

A: Factoring a trinomial involves expressing it as a product of two binomials, while simplifying a trinomial involves combining like terms to get a simpler expression.

Q: Can I factor a trinomial with a zero coefficient?

A: No, you cannot factor a trinomial with a zero coefficient. If the coefficient of the quadratic term is zero, then the trinomial is not a quadratic expression, and it cannot be factored.

Q: How do I know if a trinomial can be factored?

A: To determine if a trinomial can be factored, you need to look at the coefficients and see if they can be expressed as a product of two binomials. If they can, then the trinomial can be factored.

Tips and Tricks

Tip 1

When factoring a trinomial, always start by identifying the coefficients.

Tip 2

When determining the signs of the binomials, remember that if the product of the two binomials is positive, then both binomials must have the same sign.

Tip 3

When finding the factors of the constant term, look for two numbers whose product equals the constant term and whose sum equals the coefficient of the linear term.

Tip 4

When writing the factored form, make sure to include the correct signs of the binomials.

Common Mistakes

Mistake 1

Not identifying the coefficients correctly.

Mistake 2

Not determining the signs of the binomials correctly.

Mistake 3

Not finding the factors of the constant term correctly.

Mistake 4

Not writing the factored form correctly.

Conclusion

Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we have answered some frequently asked questions about factoring trinomials and provided tips and tricks to help you master the factoring process.