Factor The Trinomial:$3x^2 + 16x + 21$

Feb 26, 2025 by ADMIN 39 views

**Factoring Trinomials: A Step-by-Step Guide**

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Introduction

Factoring trinomials 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 $3x^2 + 16x + 21$ . Factoring trinomials is an essential skill that can be applied to a wide range of mathematical problems, from solving quadratic equations to graphing functions.

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 case of the trinomial $3x^2 + 16x + 21$ , we have $a = 3$ , $b = 16$ , and $c = 21$ .

The Factoring Process

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

Identify the coefficients: Identify the coefficients of the trinomial, which are the numbers that multiply the variable $x$ . In this case, the coefficients are $3$ , $16$ , and $21$ .
Look for common factors: Look for common factors among the coefficients. In this case, there are no common factors.
Use the factoring formula: Use the factoring formula to factor the trinomial. The factoring formula is:

ax^2 + bx + c = (mx + n)(px + q)

where $m$ , $n$ , $p$ , and $q$ are constants.

Factoring the Trinomial $3x^2 + 16x + 21$

To factor the trinomial $3x^2 + 16x + 21$ , we can use the factoring formula. We need to find four constants $m$ , $n$ , $p$ , and $q$ such that:

3x^2 + 16x + 21 = (mx + n)(px + q)

We can start by looking for two numbers whose product is $21$ and whose sum is $16$ . These numbers are $7$ and $3$ , since $7 \times 3 = 21$ and $7 + 3 = 10$ , but we need a sum of 16, so we will try other combinations.

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Introduction

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will provide a Q&A guide to help you understand the concept of factoring trinomials and how to apply it to solve problems.

Q: 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.

Q: How do I factor a trinomial?

To factor a trinomial, you need to find two binomials whose product is equal to the original trinomial. The factoring process can be broken down into several steps:

Identify the coefficients: Identify the coefficients of the trinomial, which are the numbers that multiply the variable $x$ .
Look for common factors: Look for common factors among the coefficients.
Use the factoring formula: Use the factoring formula to factor the trinomial.

Q: What is the factoring formula?

The factoring formula is:

ax^2 + bx + c = (mx + n)(px + q)

where $m$ , $n$ , $p$ , and $q$ are constants.

Q: How do I find the values of $m$ , $n$ , $p$ , and $q$ ?

To find the values of $m$ , $n$ , $p$ , and $q$ , you need to look for two numbers whose product is $ac$ and whose sum is $b$ . These numbers are the values of $m$ and $p$ . The values of $n$ and $q$ are the values of $m$ and $p$ multiplied by $x$ .

Q: What if I don't find two numbers whose product is $ac$ and whose sum is $b$ ?

If you don't find two numbers whose product is $ac$ and whose sum is $b$ , then the trinomial cannot be factored using the factoring formula.

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

Yes, you can factor a trinomial with a negative coefficient. To do this, you need to multiply the entire trinomial by $-1$ before factoring.

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

Yes, you can factor a trinomial with a zero coefficient. To do this, you need to set the coefficient of the $x^2$ term to zero and then factor the resulting trinomial.

Q: What if I make a mistake while factoring a trinomial?

If you make a mistake while factoring a trinomial, you can try to identify the error and correct it. If you are still having trouble, you can try to use a different method to factor the trinomial.

Conclusion

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. By following the steps outlined in this Q&A guide, you can learn how to factor trinomials and apply this skill to solve problems.

Common Mistakes to Avoid

When factoring trinomials, there are several common mistakes to avoid:

Not identifying the coefficients: Make sure to identify the coefficients of the trinomial before factoring.
Not looking for common factors: Make sure to look for common factors among the coefficients before factoring.
Not using the factoring formula: Make sure to use the factoring formula to factor the trinomial.
Not finding the values of $m$ , $n$ , $p$ , and $q$ : Make sure to find the values of $m$ , $n$ , $p$ , and $q$ before factoring.
Not checking the work: Make sure to check the work to ensure that the trinomial has been factored correctly.

Practice Problems

To practice factoring trinomials, try the following problems:

Problem 1: Factor the trinomial $x^2 + 5x + 6$ .
Problem 2: Factor the trinomial $x^2 - 3x - 4$ .
Problem 3: Factor the trinomial $x^2 + 2x - 15$ .

Answer Key

Problem 1: $(x + 2)(x + 3)$
Problem 2: $(x - 4)(x + 1)$
Problem 3: $(x + 5)(x - 3)$

Introduction

What is a Trinomial?

The Factoring Process

Factoring the Trinomial 3x2+16x+213x^2 + 16x + 213x2+16x+21

Introduction

Q: What is a trinomial?

Q: How do I factor a trinomial?

Q: What is the factoring formula?

Q: How do I find the values of mmm, nnn, ppp, and qqq?

Q: What if I don't find two numbers whose product is acacac and whose sum is bbb?

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

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

Q: What if I make a mistake while factoring a trinomial?

Conclusion

Common Mistakes to Avoid

Practice Problems

Answer Key

Factoring the Trinomial $3x^2 + 16x + 21$

Q: How do I find the values of $m$ , $n$ , $p$ , and $q$ ?

Q: What if I don't find two numbers whose product is $ac$ and whose sum is $b$ ?