Which Of The Binomials Below Is A Factor Of This Trinomial?$3x^2 + 18x - 21$A. $x+3$ B. $x+1$ C. $x-1$ D. $x-3$

Introduction

In algebra, factoring trinomials is a crucial skill that helps us simplify complex expressions and solve equations. A trinomial is a polynomial with three terms, and factoring it involves expressing it as a product of two binomials. In this article, we will explore the process of factoring trinomials and apply it to a specific problem.

What is a Trinomial?

A trinomial is a polynomial with three terms, which can be added, subtracted, multiplied, or divided. It is typically written in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. For example, $3x^2 + 18x - 21$ is a trinomial.

The Process of Factoring Trinomials

Factoring a trinomial involves expressing it as a product of two binomials. The process involves the following steps:

1. Identify the coefficients: Identify the coefficients of the trinomial, which are the numbers in front of the variable $x$. In the given trinomial $3x^2 + 18x - 21$, the coefficients are $3$, $18$, and $-21$.
2. Find the greatest common factor (GCF): Find the greatest common factor of the coefficients. In this case, the GCF of $3$, $18$, and $-21$ is $3$.
3. Factor out the GCF: Factor out the GCF from each term in the trinomial. In this case, we can factor out $3$ from each term to get $3(x^2 + 6x - 7)$.
4. Identify the quadratic expression: Identify the quadratic expression inside the parentheses. In this case, the quadratic expression is $x^2 + 6x - 7$.
5. Factor the quadratic expression: Factor the quadratic expression using the factoring method. In this case, we can factor the quadratic expression as $(x + 7)(x - 1)$.

Factoring the Given Trinomial

Now that we have understood the process of factoring trinomials, let's apply it to the given trinomial $3x^2 + 18x - 21$. We can factor out the GCF of $3$ from each term to get $3(x^2 + 6x - 7)$. The quadratic expression inside the parentheses is $x^2 + 6x - 7$. We can factor this quadratic expression as $(x + 7)(x - 1)$.

Which of the Binomials is a Factor of the Trinomial?

Now that we have factored the trinomial, let's examine the options given in the problem. The options are:

A. $x+3$ B. $x+1$ C. $x-1$ D. $x-3$

We can see that the factored form of the trinomial is $3(x + 7)(x - 1)$. This means that the binomial $x - 1$ is a factor of the trinomial.

Conclusion

In conclusion, factoring trinomials involves expressing them as a product of two binomials. We can factor out the greatest common factor from each term and then factor the quadratic expression inside the parentheses. In this article, we applied this process to the given trinomial $3x^2 + 18x - 21$ and found that the binomial $x - 1$ is a factor of the trinomial.

Final Answer

Introduction

In our previous article, we explored the process of factoring trinomials and applied it to a specific problem. In this article, we will answer some frequently asked questions about factoring trinomials.

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 reduce the expression to its simplest form.

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

A: To determine if a trinomial can be factored, you need to check if it can be expressed as a product of two binomials. You can do this by looking for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term.

Q: What is the greatest common factor (GCF) and how do I find it?

A: The greatest common factor (GCF) is the largest number that divides each term of a polynomial without leaving a remainder. To find the GCF, you can list the factors of each term and find the greatest common factor.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term. You can then write the quadratic expression as a product of two binomials.

Q: What is the difference between factoring by grouping and factoring by splitting?

A: Factoring by grouping involves factoring a quadratic expression by grouping the terms into pairs and factoring each pair separately. Factoring by splitting involves factoring a quadratic expression by splitting it into two binomials.

Q: Can a trinomial have more than two factors?

A: Yes, a trinomial can have more than two factors. For example, the trinomial $x^2 + 4x + 4$ can be factored as $(x + 2)(x + 2)$, which has two identical factors.

Q: How do I check if a factored form of a trinomial is correct?

A: To check if a factored form of a trinomial is correct, you can multiply the two binomials together and see if you get the original trinomial.

Q: Can a trinomial be factored if it has a negative coefficient?

A: Yes, a trinomial can be factored even if it has a negative coefficient. For example, the trinomial $-x^2 - 6x - 8$ can be factored as $-(x + 4)(x + 2)$.

Q: How do I factor a trinomial with a variable coefficient?

A: To factor a trinomial with a variable coefficient, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term. You can then write the trinomial as a product of two binomials.

Conclusion

In conclusion, factoring trinomials involves expressing them as a product of two binomials. We have answered some frequently asked questions about factoring trinomials and provided examples to illustrate the concepts.

Final Answer

The final answer is that factoring trinomials is a crucial skill in algebra that helps us simplify complex expressions and solve equations. By understanding the process of factoring trinomials, we can apply it to a wide range of problems and become proficient in algebra.