Comparing the Factored Forms

Oct 13, 2025 by windailys.com 36 views

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will explore the process of factoring polynomials, with a focus on the given polynomial: $3x^3 - 15x^2 + 8x - 40$ . We will also examine the two possible factored forms of the polynomial and determine which one is the completely factored form.

The Given Polynomial

The given polynomial is:

3x^3 - 15x^2 + 8x - 40

This polynomial has four terms, and we are asked to factor it.

Factoring by Grouping

One way to factor the polynomial is by grouping. We can group the first two terms and the last two terms:

3x^3 - 15x^2 + 8x - 40 = (3x^3 - 15x^2) + (8x - 40)

Now, we can factor out the greatest common factor (GCF) from each group:

3x^3 - 15x^2 + 8x - 40 = 3x^2(x - 5) + 8(x - 5)

Factoring the Polynomial

Now that we have factored the polynomial by grouping, we can see that both terms have a common factor of $(x - 5)$ . We can factor this out:

3x^3 - 15x^2 + 8x - 40 = (3x^2 + 8)(x - 5)

Comparing the Factored Forms

We are given two possible factored forms of the polynomial:

A. $(3x^2 - 5)(x + 8)$ B. $(3x^2 - 8)(x + 5)$

We need to determine which one is the completely factored form.

Analyzing Option A

Let's analyze option A:

(3x^2 - 5)(x + 8)

This expression is not equal to the original polynomial. Therefore, option A is not the completely factored form.

Analyzing Option B

Now, let's analyze option B:

(3x^2 - 8)(x + 5)

This expression is not equal to the original polynomial. Therefore, option B is not the completely factored form.

Conclusion

We have factored the polynomial by grouping and found that the completely factored form is:

(3x^2 + 8)(x - 5)

This is the correct answer.

Tips and Tricks

When factoring polynomials, look for common factors and group terms accordingly.
Use the distributive property to expand expressions and simplify them.
Check your work by plugging in values or using a calculator to verify the factored form.

Common Mistakes

Failing to factor out the greatest common factor (GCF) from each group.
Not checking the work by plugging in values or using a calculator.
Not using the distributive property to expand expressions and simplify them.

Real-World Applications

Factoring polynomials has many real-world applications, including:

Solving systems of equations
Finding the roots of a polynomial
Modeling real-world phenomena using algebraic equations

Conclusion

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we explored the process of factoring polynomials and determined the completely factored form of the given polynomial. In this article, we will answer some frequently asked questions about factoring polynomials.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This means that we break down the polynomial into smaller parts, called factors, that can be multiplied together to get the original polynomial.

Q: Why is factoring a polynomial important?

A: Factoring a polynomial is important because it allows us to simplify complex expressions and solve equations more easily. It also helps us to identify the roots of a polynomial, which is essential in many real-world applications.

Q: What are some common methods for factoring polynomials?

A: Some common methods for factoring polynomials include:

Factoring by grouping
Factoring out the greatest common factor (GCF)
Using the distributive property to expand expressions and simplify them
Using the quadratic formula to factor quadratic expressions

Q: How do I factor a polynomial with multiple terms?

A: To factor a polynomial with multiple terms, follow these steps:

Look for common factors and group terms accordingly.
Factor out the greatest common factor (GCF) from each group.
Use the distributive property to expand expressions and simplify them.
Check your work by plugging in values or using a calculator to verify the factored form.

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

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, while simplifying a polynomial involves reducing it to its simplest form. For example, the polynomial $2x^2 + 4x + 2$ can be factored as $(2x + 1)(x + 2)$ , but it can also be simplified as $2(x + 1)(x + 2)$ .

Q: Can I use a calculator to factor a polynomial?

A: Yes, you can use a calculator to factor a polynomial. Many calculators have built-in functions for factoring polynomials, such as the "factor" function on a graphing calculator. However, it's always a good idea to check your work by plugging in values or using a calculator to verify the factored form.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

Failing to factor out the greatest common factor (GCF) from each group.
Not checking the work by plugging in values or using a calculator to verify the factored form.
Not using the distributive property to expand expressions and simplify them.

Q: How do I know if a polynomial is factored correctly?

A: To determine if a polynomial is factored correctly, follow these steps:

Check the work by plugging in values or using a calculator to verify the factored form.
Use the distributive property to expand expressions and simplify them.
Verify that the factored form is equivalent to the original polynomial.

Conclusion

In conclusion, factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By following the steps outlined in this article, you can factor polynomials and determine the completely factored form. Remember to look for common factors, group terms accordingly, and use the distributive property to expand expressions and simplify them. With practice and patience, you can become proficient in factoring polynomials and apply this skill to real-world problems.