Factor Completely $2x^3 + 10x^2 + 14x + 70$.A. $(2x^2 + 14)(x + 5)$ B. $ ( X 2 + 7 ) ( 2 X + 10 ) (x^2 + 7)(2x + 10) ( X 2 + 7 ) ( 2 X + 10 ) [/tex] C. $2(x^3 + 5x^2 + 7x + 35)$ D. $2[(x^2 + 7)(x + 5)]$

Factor Completely: A Step-by-Step Guide to Factoring the Polynomial $2x^3 + 10x^2 + 14x + 70$

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the polynomial $2x^3 + 10x^2 + 14x + 70$. We will explore the different methods of factoring and provide a step-by-step guide to help you understand the process.

Understanding the Polynomial

Before we begin factoring, let's take a closer look at the polynomial $2x^3 + 10x^2 + 14x + 70$. We can see that it is a cubic polynomial, meaning it has a degree of 3. The polynomial has four terms: $2x^3$, $10x^2$, $14x$, and $70$.

Factoring by Grouping

One method of factoring is by grouping. This involves grouping the terms of the polynomial into pairs and then factoring out the greatest common factor (GCF) from each pair.

Let's start by grouping the first two terms: $2x^3$ and $10x^2$. We can factor out the GCF, which is $2x^2$, from these two terms.

2x^3 + 10x^2 = 2x^2(x + 5)

Next, we can group the last two terms: $14x$ and $70$. We can factor out the GCF, which is $14$, from these two terms.

14x + 70 = 14(x + 5)

Now, we can see that both pairs have a common factor of $(x + 5)$. We can factor this out to get the final factored form.

2x^3 + 10x^2 + 14x + 70 = 2x^2(x + 5) + 14(x + 5)
= (2x^2 + 14)(x + 5)

Factoring by Greatest Common Factor (GCF)

Another method of factoring is by finding the greatest common factor (GCF) of the polynomial. The GCF is the largest factor that divides all the terms of the polynomial.

In this case, the GCF of the polynomial $2x^3 + 10x^2 + 14x + 70$ is 2. We can factor out the GCF from each term to get:

2x^3 + 10x^2 + 14x + 70 = 2(x^3 + 5x^2 + 7x + 35)

Factoring by Difference of Squares

The difference of squares is a special case of factoring that involves a polynomial of the form $a^2 - b^2$. We can factor this as $(a + b)(a - b)$.

However, the polynomial $2x^3 + 10x^2 + 14x + 70$ does not fit this form. Therefore, we cannot factor it using the difference of squares method.

Factoring by Sum and Difference

The sum and difference method involves factoring a polynomial of the form $a^2 + 2ab + b^2$ as $(a + b)^2$. However, the polynomial $2x^3 + 10x^2 + 14x + 70$ does not fit this form. Therefore, we cannot factor it using the sum and difference method.

In conclusion, we have factored the polynomial $2x^3 + 10x^2 + 14x + 70$ using the method of factoring by grouping. We have also explored other methods of factoring, including factoring by greatest common factor (GCF) and factoring by difference of squares. However, these methods do not apply to this particular polynomial.

The final answer is:

• A. $(2x^2 + 14)(x + 5)$

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Q: What is factoring in algebra?

A: Factoring in algebra involves expressing a polynomial as a product of simpler polynomials. It is a way of simplifying complex expressions and making them easier to work with.

Q: Why is factoring important in algebra?

A: Factoring is important in algebra because it allows us to simplify complex expressions and make them easier to work with. It also helps us to identify the roots of a polynomial, which is essential in many areas of mathematics and science.

Q: What are the different methods of factoring?

A: There are several methods of factoring, including:

• Factoring by grouping
• Factoring by greatest common factor (GCF)
• Factoring by difference of squares
• Factoring by sum and difference

Q: How do I factor a polynomial using the method of factoring by grouping?

A: To factor a polynomial using the method of factoring by grouping, you need to group the terms of the polynomial into pairs and then factor out the greatest common factor (GCF) from each pair.

Q: How do I factor a polynomial using the method of factoring by greatest common factor (GCF)?

A: To factor a polynomial using the method of factoring by greatest common factor (GCF), you need to find the greatest common factor (GCF) of the polynomial and then factor it out from each term.

Q: What is the difference of squares method of factoring?

A: The difference of squares method of factoring involves factoring a polynomial of the form $a^2 - b^2$ as $(a + b)(a - b)$.

Q: Can you give an example of factoring a polynomial using the difference of squares method?

A: Yes, here is an example of factoring a polynomial using the difference of squares method:

$$x^2 - 9 = (x + 3)(x - 3)$$

Q: What is the sum and difference method of factoring?

A: The sum and difference method of factoring involves factoring a polynomial of the form $a^2 + 2ab + b^2$ as $(a + b)^2$.

Q: Can you give an example of factoring a polynomial using the sum and difference method?

A: Yes, here is an example of factoring a polynomial using the sum and difference method:

$$x^2 + 6x + 9 = (x + 3)^2$$

Q: How do I know which method of factoring to use?

A: To determine which method of factoring to use, you need to examine the polynomial and look for patterns or relationships between the terms. You can also try using different methods and see which one works best.

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

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

• Not factoring out the greatest common factor (GCF) from each term
• Not grouping the terms correctly
• Not recognizing the difference of squares or sum and difference patterns
• Not checking for common factors between terms

Q: How can I practice factoring polynomials?

A: You can practice factoring polynomials by working through examples and exercises in your textbook or online resources. You can also try factoring polynomials on your own and then checking your work with a calculator or online tool.

In conclusion, factoring polynomials is an important skill in algebra that involves expressing a polynomial as a product of simpler polynomials. There are several methods of factoring, including factoring by grouping, factoring by greatest common factor (GCF), factoring by difference of squares, and factoring by sum and difference. By understanding these methods and practicing factoring polynomials, you can become proficient in this skill and apply it to a wide range of mathematical and scientific problems.