Which Of The Binomials Below Is A Factor Of This Trinomial?$5x^2 - 5x - 100$A. $x - 5$ B. $x + 10$ C. $x - 4$ D. $x + 5$

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, $5x^2 - 5x - 100$ is a trinomial.

The Process of Factoring Trinomials

Factoring trinomials involves expressing it as a product of two binomials. The process involves finding two numbers whose product is equal to the product of the coefficient of the $x^2$ term and the constant term, and whose sum is equal to the coefficient of the $x$ term. These two numbers are called the "factors" of the trinomial.

Step 1: Identify the Coefficients

To factor a trinomial, we need to identify the coefficients of the $x^2$ term, the $x$ term, and the constant term. In the trinomial $5x^2 - 5x - 100$, the coefficients are:

• Coefficient of $x^2$: 5
• Coefficient of $x$: -5
• Constant term: -100

Step 2: Find the Factors

Next, we need to find two numbers whose product is equal to the product of the coefficient of the $x^2$ term and the constant term, and whose sum is equal to the coefficient of the $x$ term. In this case, we need to find two numbers whose product is equal to $5 \times -100 = -500$ and whose sum is equal to $-5$.

Step 3: Factor the Trinomial

Once we have found the two numbers, we can factor the trinomial by expressing it as a product of two binomials. In this case, we can factor the trinomial as follows:

$5x^2 - 5x - 100 = (5x + 10)(x - 10)$

Which of the Binomials is a Factor of the Trinomial?

Now that we have factored the trinomial, we can see that the binomial $5x + 10$ is a factor of the trinomial. However, we are given four options: $x - 5$, $x + 10$, $x - 4$, and $x + 5$. Which one of these binomials is a factor of the trinomial?

Analyzing the Options

Let's analyze each option:

• Option A: $x - 5$. This binomial is not a factor of the trinomial, as it does not divide the trinomial evenly.
• Option B: $x + 10$. This binomial is a factor of the trinomial, as it divides the trinomial evenly.
• Option C: $x - 4$. This binomial is not a factor of the trinomial, as it does not divide the trinomial evenly.
• Option D: $x + 5$. This binomial is not a factor of the trinomial, as it does not divide the trinomial evenly.

Conclusion

In conclusion, the binomial $x + 10$ is a factor of the trinomial $5x^2 - 5x - 100$. This is because it divides the trinomial evenly, and it is one of the factors of the trinomial.

Final Answer

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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 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.

Q: How do I factor a trinomial?

To factor a trinomial, you need to follow these steps:

1. Identify the coefficients of the $x^2$ term, the $x$ term, and the constant term.
2. Find two numbers whose product is equal to the product of the coefficient of the $x^2$ term and the constant term, and whose sum is equal to the coefficient of the $x$ term.
3. Factor the trinomial by expressing it as a product of two binomials.

Q: What are the common mistakes to avoid when factoring trinomials?

Some common mistakes to avoid when factoring trinomials include:

• Not identifying the coefficients correctly
• Not finding the correct factors
• Not factoring the trinomial correctly
• Not checking if the factors are correct

Q: How do I check if the factors are correct?

To check if the factors are correct, you can multiply the two binomials together and see if you get the original trinomial. If you do, then the factors are correct.

Q: What are some tips for factoring trinomials?

Some tips for factoring trinomials include:

• Use the distributive property to expand the trinomial
• Look for common factors
• Use the factoring method of grouping
• Check your work carefully

Q: Can I factor a trinomial that has a negative coefficient?

Yes, you can factor a trinomial that has a negative coefficient. The process is the same as factoring a trinomial with a positive coefficient.

Q: Can I factor a trinomial that has a variable coefficient?

Yes, you can factor a trinomial that has a variable coefficient. The process is the same as factoring a trinomial with a constant coefficient.

Q: How do I factor a trinomial with a quadratic expression?

To factor a trinomial with a quadratic expression, you need to follow the same steps as factoring a trinomial with a constant coefficient. However, you may need to use the quadratic formula to find the roots of the quadratic expression.

Q: Can I factor a trinomial that has a complex coefficient?

Yes, you can factor a trinomial that has a complex coefficient. The process is the same as factoring a trinomial with a real coefficient.

Conclusion

In conclusion, factoring trinomials is a crucial skill that helps us simplify complex expressions and solve equations. By following the steps outlined in this article, you can factor trinomials with ease. Remember to check your work carefully and use the tips and tricks outlined in this article to help you factor trinomials like a pro.

Final Answer

The final answer is: There is no final answer, as this is a Q&A article.