Applying Factoring to the Given Problem

Oct 13, 2025 by magicalceylontours.com 20 views

Introduction

In algebra, factoring binomials is a crucial concept that helps us simplify complex expressions and solve equations. When given a quadratic expression, we need to identify the factors that multiply together to give us the original expression. In this article, we will explore how to factor binomials and apply this concept to solve a specific problem.

What are Binomials?

A binomial is an algebraic expression consisting of two terms. It can be written in the form of $ax + b$ , where $a$ and $b$ are constants, and $x$ is the variable. For example, $3x + 4$ and $2x - 5$ are both binomials.

Factoring Binomials

To factor a binomial, we need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the variable term. Let's consider an example to illustrate this concept.

Suppose we want to factor the binomial $6x + 8$ . We need to find two numbers whose product is equal to $8$ and whose sum is equal to $6$ . By trial and error, we can find that $4$ and $2$ satisfy these conditions, since $4 \times 2 = 8$ and $4 + 2 = 6$ . Therefore, we can write $6x + 8$ as $(4x + 2)(x + 4)$ .

Applying Factoring to the Given Problem

Now, let's apply the concept of factoring binomials to the given problem. We are asked to select the two binomials that are factors of $49x^2 - 36$ . To do this, we need to find two binomials whose product is equal to $49x^2 - 36$ .

Step 1: Factor the Constant Term

The constant term in the given expression is $-36$ . We can factor this as $-36 = -1 \times 36 = 1 \times -36 = 6 \times -6 = 9 \times -4 = 12 \times -3$ .

Step 2: Find the Factors of the Variable Term

The variable term in the given expression is $49x^2$ . We can factor this as $49x^2 = 7x \times 7x = 7x \times 1 \times 7x = 7x \times 7x \times 1$ .

Step 3: Combine the Factors

Now, we need to combine the factors of the constant term and the variable term to form two binomials. We can do this by multiplying the factors of the constant term with the factors of the variable term.

Let's consider the factors of the constant term: $-1 \times 36$ , $1 \times -36$ , $6 \times -6$ , $9 \times -4$ , and $12 \times -3$ . We can pair these factors with the factors of the variable term: $7x \times 7x$ , $7x \times 1 \times 7x$ , and $7x \times 7x \times 1$ .

By multiplying these pairs, we get the following binomials:

$-1 \times 36 \times 7x \times 7x = -1 \times 36 \times 7x^2 = -252x^2$
$1 \times -36 \times 7x \times 7x = 1 \times -36 \times 7x^2 = -252x^2$
$6 \times -6 \times 7x \times 7x = 6 \times -6 \times 7x^2 = -252x^2$
$9 \times -4 \times 7x \times 7x = 9 \times -4 \times 7x^2 = -252x^2$
$12 \times -3 \times 7x \times 7x = 12 \times -3 \times 7x^2 = -252x^2$

However, we are looking for binomials that multiply together to give us the original expression $49x^2 - 36$ . We can see that none of the above binomials satisfy this condition.

Step 4: Check the Answer Choices

Let's go back to the answer choices and see if any of them satisfy the condition. We have the following options:

A. $7x + 12$ B. $7x - 6$ C. $7x - 3$ D. $7x + 6$

We can multiply each of these binomials with its corresponding binomial to see if we get the original expression.

$(7x + 12)(7x - 12) = 49x^2 - 144 \neq 49x^2 - 36$
$(7x - 6)(7x + 6) = 49x^2 - 36$
$(7x - 3)(7x + 3) = 49x^2 - 9 \neq 49x^2 - 36$
$(7x + 6)(7x - 6) = 49x^2 - 36$

We can see that options B and D satisfy the condition.

Conclusion

In conclusion, the two binomials that are factors of $49x^2 - 36$ are $7x - 6$ and $7x + 6$ . These binomials multiply together to give us the original expression.

Final Answer

The final answer is:

Introduction

In our previous article, we explored how to factor binomials and applied this concept to solve a specific problem. In this article, we will provide a Q&A guide to help you better understand the concept of factoring binomials.

Q: What is a binomial?

A: A binomial is an algebraic expression consisting of two terms. It can be written in the form of $ax + b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I factor a binomial?

A: To factor a binomial, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the variable term.

Q: What are the steps to factor a binomial?

A: The steps to factor a binomial are:

Factor the constant term.
Find the factors of the variable term.
Combine the factors to form two binomials.

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

A: To find the factors of the constant term, you need to list all the possible pairs of numbers that multiply together to give you the constant term.

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

A: To find the factors of the variable term, you need to list all the possible pairs of numbers that multiply together to give you the variable term.

Q: What if I have a quadratic expression with a negative sign?

A: If you have a quadratic expression with a negative sign, you can factor it by finding two binomials whose product is equal to the original expression.

Q: Can I factor a binomial with a variable in the constant term?

A: No, you cannot factor a binomial with a variable in the constant term. This is because the constant term must be a constant value, not a variable.

Q: Can I factor a binomial with a variable in the coefficient of the variable term?

A: Yes, you can factor a binomial with a variable in the coefficient of the variable term. However, you need to be careful when factoring, as the variable may be raised to a power.

Q: What if I have a binomial with a coefficient of 1?

A: If you have a binomial with a coefficient of 1, you can factor it by simply listing the two terms.

Q: Can I factor a binomial with a coefficient of -1?

A: Yes, you can factor a binomial with a coefficient of -1. However, you need to be careful when factoring, as the negative sign may affect the signs of the factors.

Q: What if I have a binomial with a variable raised to a power?

A: If you have a binomial with a variable raised to a power, you can factor it by finding two binomials whose product is equal to the original expression.

Q: Can I factor a binomial with a variable raised to a power and a coefficient?

A: Yes, you can factor a binomial with a variable raised to a power and a coefficient. However, you need to be careful when factoring, as the variable may be raised to a power and the coefficient may affect the signs of the factors.

Conclusion

In conclusion, factoring binomials is a crucial concept in algebra that helps us simplify complex expressions and solve equations. By following the steps outlined in this article, you can factor binomials with ease and apply this concept to solve a variety of problems.

Final Tips

Always start by factoring the constant term.
Find the factors of the variable term by listing all the possible pairs of numbers that multiply together to give you the variable term.
Combine the factors to form two binomials.
Be careful when factoring binomials with variables raised to powers and coefficients.
Practice, practice, practice! The more you practice factoring binomials, the more comfortable you will become with the concept.

Common Mistakes to Avoid

Factoring a binomial with a variable in the constant term.
Factoring a binomial with a variable raised to a power and a coefficient without considering the signs of the factors.
Not considering the coefficient of the variable term when factoring.
Not listing all the possible pairs of numbers that multiply together to give you the variable term.

Additional Resources

Khan Academy: Factoring Binomials
Mathway: Factoring Binomials
Wolfram Alpha: Factoring Binomials

Conclusion

In conclusion, factoring binomials is a crucial concept in algebra that helps us simplify complex expressions and solve equations. By following the steps outlined in this article and avoiding common mistakes, you can factor binomials with ease and apply this concept to solve a variety of problems.