Factor Completely.${ 2x^3 + 16x^2 - 18x }$1. ${ 2x(x^2 + 8x - 9) }$2. ${ 2x(x + 3)(x - 3) }$3. ${ 2x(x + 1)(x - 9) }$4. ${ 2x(x - 1)(x + 9) }$

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 focus on factoring the given polynomial $2x^3 + 16x^2 - 18x$ completely. We will explore different methods and techniques to factor the polynomial and provide step-by-step solutions.

Understanding the Polynomial

Before we begin factoring, let's take a closer look at the given polynomial $2x^3 + 16x^2 - 18x$. We can see that it is a cubic polynomial, meaning it has three terms. The first term is $2x^3$, the second term is $16x^2$, and the third term is $-18x$.

Factoring by Grouping

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

Step 1: Group the terms

We can group the first two terms together and the third term separately.

$2x^3 + 16x^2 - 18x = (2x^3 + 16x^2) - 18x$

Step 2: Factor out the GCF from each pair

Now, we can factor out the GCF from each pair.

$(2x^3 + 16x^2) - 18x = 2x^2(x + 8) - 18x$

Step 3: Factor out the common factor

We can see that both terms have a common factor of $2x$. We can factor this out.

$2x^2(x + 8) - 18x = 2x(x^2 + 8x - 9)$

Step 4: Factor the quadratic expression

Now, we can factor the quadratic expression $x^2 + 8x - 9$.

$x^2 + 8x - 9 = (x + 9)(x - 1)$

Step 5: Write the final factored form

We can now write the final factored form of the polynomial.

$2x^3 + 16x^2 - 18x = 2x(x + 9)(x - 1)$

Alternative Factoring Methods

There are other methods to factor the polynomial, such as factoring by grouping or using the quadratic formula. However, the method we used above is a common and efficient way to factor the polynomial.

Factoring by Grouping (Alternative)

We can also factor the polynomial by grouping the terms in a different way.

$2x^3 + 16x^2 - 18x = (2x^3 + 16x^2) - 18x$

$= 2x^2(x + 8) - 18x$

$= 2x(x^2 + 8x - 9)$

$= 2x(x + 9)(x - 1)$

Using the Quadratic Formula

We can also use the quadratic formula to factor the polynomial. However, this method is more complex and requires a deeper understanding of algebra.

Conclusion

In this article, we have explored different methods to factor the polynomial $2x^3 + 16x^2 - 18x$ completely. We have used the method of factoring by grouping and provided step-by-step solutions. We have also discussed alternative factoring methods, such as factoring by grouping and using the quadratic formula. By understanding and applying these methods, we can factor polynomials efficiently and effectively.

Final Answer

The final factored form of the polynomial is:

$2x^3 + 16x^2 - 18x = 2x(x + 9)(x - 1)$

Discussion

This problem is a great example of how to factor polynomials completely. The method of factoring by grouping is a common and efficient way to factor polynomials. By understanding and applying this method, we can factor polynomials efficiently and effectively. We can also use alternative factoring methods, such as factoring by grouping and using the quadratic formula, to factor polynomials.

Common Mistakes

When factoring polynomials, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not factoring out the greatest common factor (GCF) from each pair of terms.
• Not factoring the quadratic expression completely.
• Not writing the final factored form of the polynomial correctly.

Tips and Tricks

Here are some tips and tricks to help you factor polynomials efficiently and effectively:

• Use the method of factoring by grouping to factor polynomials.
• Factor out the greatest common factor (GCF) from each pair of terms.
• Factor the quadratic expression completely.
• Write the final factored form of the polynomial correctly.

Practice Problems

Here are some practice problems to help you practice factoring polynomials:

• Factor the polynomial $x^3 + 8x^2 - 9x$ completely.
• Factor the polynomial $2x^3 + 16x^2 - 18x$ completely.
• Factor the polynomial $x^3 + 8x^2 - 9x$ using the quadratic formula.

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 different methods to factor the polynomial $2x^3 + 16x^2 - 18x$ completely. In this article, we will provide a Q&A guide to help you understand and apply the concepts of 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 can break down a 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 helps us to:

• Simplify complex polynomials
• Solve equations and inequalities
• Find the roots of a polynomial
• Understand the behavior of a polynomial

Q: What are the different methods of factoring a polynomial?

A: There are several methods of factoring a polynomial, including:

• Factoring by grouping
• Factoring by using the quadratic formula
• Factoring by using the difference of squares
• Factoring by using the sum and difference of cubes

Q: How do I factor a polynomial by grouping?

A: To factor a polynomial by grouping, follow these steps:

1. Group the terms of the polynomial into pairs.
2. Factor out the greatest common factor (GCF) from each pair.
3. Factor out the common factor from the remaining terms.
4. Write the final factored form of the polynomial.

Q: How do I factor a polynomial using the quadratic formula?

A: To factor a polynomial using the quadratic formula, follow these steps:

1. Write the polynomial in the form $ax^2 + bx + c$.
2. Plug the values of $a$, $b$, and $c$ into the quadratic formula.
3. Simplify the expression to get the factored form of the polynomial.

Q: What is the difference of squares?

A: The difference of squares is a special case of factoring a polynomial. It involves factoring a polynomial of the form $a^2 - b^2$.

Q: How do I factor a polynomial using the difference of squares?

A: To factor a polynomial using the difference of squares, follow these steps:

1. Identify the polynomial as a difference of squares.
2. Factor the polynomial into the form $(a + b)(a - b)$.

Q: What is the sum and difference of cubes?

A: The sum and difference of cubes are special cases of factoring a polynomial. They involve factoring polynomials of the form $a^3 + b^3$ and $a^3 - b^3$.

Q: How do I factor a polynomial using the sum and difference of cubes?

A: To factor a polynomial using the sum and difference of cubes, follow these steps:

1. Identify the polynomial as a sum or difference of cubes.
2. Factor the polynomial into the form $(a + b)(a^2 - ab + b^2)$ or $(a - b)(a^2 + ab + b^2)$.

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

A: Some common mistakes to avoid when factoring a polynomial include:

• Not factoring out the greatest common factor (GCF) from each pair of terms.
• Not factoring the quadratic expression completely.
• Not writing the final factored form of the polynomial correctly.

Q: How can I practice factoring polynomials?

A: You can practice factoring polynomials by:

• Working through example problems
• Using online resources and practice tests
• Asking a teacher or tutor for help
• Joining a study group or math club

Conclusion

In conclusion, factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By understanding and applying the different methods of factoring a polynomial, you can simplify complex polynomials, solve equations and inequalities, find the roots of a polynomial, and understand the behavior of a polynomial. Remember to practice factoring polynomials regularly to become proficient in this skill.