Factor − 15 D 3 + 21 D 2 − 6 D -15d^3 + 21d^2 - 6d − 15 D 3 + 21 D 2 − 6 D Completely.

$$

Introduction

Factoring polynomials is a crucial concept in algebra, and it plays a significant role in solving various mathematical problems. In this article, we will focus on factoring the given polynomial $-15d^3 + 21d^2 - 6d$ completely. Factoring a polynomial involves expressing it as a product of simpler polynomials, known as factors. This process can be used to simplify complex expressions, solve equations, and identify the roots of a polynomial.

Understanding the Polynomial

Before we proceed with factoring the polynomial, let's analyze its structure. The given polynomial is a cubic polynomial, which means it has a degree of 3. It consists of three terms: $-15d^3$, $21d^2$, and $-6d$. The first term has a coefficient of $-15$, the second term has a coefficient of $21$, and the third term has a coefficient of $-6$. The variable $d$ is raised to the power of 3 in the first term, to the power of 2 in the second term, and to the power of 1 in the third term.

Factoring Out the Greatest Common Factor (GCF)

One of the first steps in factoring a polynomial is to identify and factor out the greatest common factor (GCF). The GCF is the largest expression that divides each term of the polynomial without leaving a remainder. In this case, the GCF of the polynomial $-15d^3 + 21d^2 - 6d$ is $-3d$. We can factor out $-3d$ from each term of the polynomial:

$$-15d^3 + 21d^2 - 6d = -3d(5d^2 - 7d + 2)$$

Factoring the Quadratic Expression

The expression inside the parentheses, $5d^2 - 7d + 2$, is a quadratic expression. We can factor this expression by finding two numbers whose product is $5 \times 2 = 10$ and whose sum is $-7$. These numbers are $-5$ and $-2$. We can rewrite the quadratic expression as:

$$5d^2 - 7d + 2 = 5d^2 - 5d - 2d + 2$$

Now, we can factor out the common terms:

$$5d^2 - 5d - 2d + 2 = 5d(d - 1) - 2(d - 1)$$

We can see that both terms have a common factor of $(d - 1)$. We can factor this out:

$$5d(d - 1) - 2(d - 1) = (5d - 2)(d - 1)$$

Factoring the Polynomial Completely

Now that we have factored the quadratic expression, we can rewrite the original polynomial as:

$$-15d^3 + 21d^2 - 6d = -3d(5d^2 - 7d + 2) = -3d(5d - 2)(d - 1)$$

This is the complete factorization of the polynomial $-15d^3 + 21d^2 - 6d$.

Conclusion

Factoring polynomials is an essential skill in algebra, and it requires a deep understanding of the structure of polynomials. In this article, we have factored the polynomial $-15d^3 + 21d^2 - 6d$ completely. We have identified the greatest common factor (GCF) and factored it out, and then we have factored the quadratic expression inside the parentheses. The complete factorization of the polynomial is $-3d(5d - 2)(d - 1)$. This factorization can be used to simplify complex expressions, solve equations, and identify the roots of the polynomial.

Applications of Factoring

Factoring polynomials has numerous applications in mathematics and other fields. Some of the applications of factoring include:

• Solving equations: Factoring polynomials can be used to solve equations by setting each factor equal to zero and solving for the variable.
• Identifying roots: Factoring polynomials can be used to identify the roots of a polynomial, which are the values of the variable that make the polynomial equal to zero.
• Simplifying expressions: Factoring polynomials can be used to simplify complex expressions by combining like terms.
• Graphing functions: Factoring polynomials can be used to graph functions by identifying the x-intercepts of the function.

Real-World Applications

Factoring polynomials has numerous real-world applications. Some of the real-world applications of factoring include:

• Physics: Factoring polynomials is used to solve problems in physics, such as finding the trajectory of an object under the influence of gravity.
• Engineering: Factoring polynomials is used to solve problems in engineering, such as designing electrical circuits and mechanical systems.
• Computer Science: Factoring polynomials is used to solve problems in computer science, such as coding theory and cryptography.
• Economics: Factoring polynomials is used to solve problems in economics, such as modeling economic systems and predicting economic trends.

Final Thoughts

$$$$

Frequently Asked Questions

Factoring polynomials can be a challenging task, especially for those who are new to algebra. In this article, we will answer some of the most 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, known as factors. This process can be used to simplify complex expressions, solve equations, and identify the roots of a polynomial.

Q: Why is factoring a polynomial important?

A: Factoring a polynomial is important because it can be used to solve equations, identify the roots of a polynomial, and simplify complex expressions. It is also a crucial concept in algebra and has numerous applications in mathematics and other fields.

Q: How do I factor a polynomial?

A: Factoring a polynomial involves identifying the greatest common factor (GCF) and factoring it out, and then factoring the remaining expression. You can also use various factoring techniques, such as factoring by grouping, factoring by difference of squares, and factoring by sum and difference.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest expression that divides each term of the polynomial without leaving a remainder. It is the first step in factoring a polynomial.

Q: How do I find the GCF of a polynomial?

A: To find the GCF of a polynomial, you need to identify the common factors of each term. You can do this by listing the factors of each term and finding the greatest common factor.

Q: What are some common factoring techniques?

A: Some common factoring techniques include:

• Factoring by grouping: This involves grouping the terms of the polynomial into pairs and factoring out the common factors.
• Factoring by difference of squares: This involves factoring the difference of squares, which is a polynomial of the form $a^2 - b^2$.
• Factoring by sum and difference: This involves factoring the sum and difference of two or more terms.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. You can then rewrite the quadratic expression as the product of two binomials.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to reduce the complexity of the expression.

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

A: Yes, you can use a calculator to factor a polynomial. However, it is always a good idea to check your work by factoring the polynomial by hand.

Q: How do I know if a polynomial is factorable?

A: A polynomial is factorable if it can be expressed as a product of simpler polynomials. You can use various factoring techniques to determine if a polynomial is factorable.

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

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

• Not identifying the GCF: Failing to identify the greatest common factor (GCF) can make it difficult to factor the polynomial.
• Not using the correct factoring technique: Using the wrong factoring technique can lead to incorrect results.
• Not checking your work: Failing to check your work can lead to errors in the factored form of the polynomial.

Conclusion

Factoring polynomials is a crucial concept in algebra, and it has numerous applications in mathematics and other fields. In this article, we have answered some of the most frequently asked questions about factoring polynomials. We hope that this article has provided you with a better understanding of factoring polynomials and has helped you to avoid common mistakes.