Find The Complete Factored Form Of The Polynomial:$-7m^6n^5 + 8n^3$

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Introduction

Factoring polynomials is a crucial concept in algebra, and it plays a vital role in solving equations and inequalities. In this article, we will focus on finding the complete factored form of the given polynomial: $-7m^6n^5 + 8n^3$. Factoring a polynomial involves expressing it as a product of simpler polynomials, known as factors. The complete factored form of a polynomial is the product of all its factors.

Understanding the Polynomial

Before we proceed with factoring the polynomial, let's understand its structure. The given polynomial is $-7m^6n^5 + 8n^3$. It consists of two terms: $-7m^6n^5$ and $8n^3$. The first term has a negative coefficient, while the second term has a positive coefficient.

Factoring Out the Greatest Common Factor (GCF)

The first step in factoring the polynomial is to identify the greatest common factor (GCF) of the two terms. The GCF is the largest expression that divides both terms without leaving a remainder. In this case, the GCF of $-7m^6n^5$ and $8n^3$ is $n^3$. We can factor out $n^3$ from both terms.

-7m^6n^5 + 8n^3 = n^3(-7m^6 + 8)

Factoring the Remaining Expression

Now that we have factored out $n^3$, we are left with the expression $-7m^6 + 8$. This expression can be factored further by identifying the greatest common factor of the two terms. In this case, the GCF is 1, which means that the expression cannot be factored further.

Factoring Out the Coefficient

However, we can factor out the coefficient of the first term, which is $-7$. We can factor out $-7$ from the expression $-7m^6 + 8$.

-7m^6 + 8 = -7(m^6 - 8/7)

Simplifying the Expression

Now that we have factored out $-7$, we can simplify the expression by rewriting it as $-7(m^6 - 8/7)$.

Factoring the Difference of Squares

The expression $m^6 - 8/7$ can be factored using the difference of squares formula. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. In this case, we can rewrite $m^6 - 8/7$ as $(m^3)^2 - (2)^2$.

m^6 - 8/7 = (m^3)^2 - (2)^2

Applying the Difference of Squares Formula

Now that we have rewritten the expression as $(m^3)^2 - (2)^2$, we can apply the difference of squares formula. The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. In this case, we can rewrite $(m^3)^2 - (2)^2$ as $(m^3 + 2)(m^3 - 2)$.

(m^3)^2 - (2)^2 = (m^3 + 2)(m^3 - 2)

Factoring the Expression

Now that we have factored the expression $(m^3)^2 - (2)^2$ as $(m^3 + 2)(m^3 - 2)$, we can rewrite the original polynomial as $-7n^3(m^3 + 2)(m^3 - 2)$.

Conclusion

In this article, we have found the complete factored form of the polynomial $-7m^6n^5 + 8n^3$. We have factored out the greatest common factor (GCF) of the two terms, which is $n^3$. We have also factored out the coefficient of the first term, which is $-7$. Finally, we have factored the remaining expression using the difference of squares formula. The complete factored form of the polynomial is $-7n^3(m^3 + 2)(m^3 - 2)$.

Final Answer

The complete factored form of the polynomial $-7m^6n^5 + 8n^3$ is $-7n^3(m^3 + 2)(m^3 - 2)$.

Introduction

Factoring polynomials is a crucial concept in algebra, and it plays a vital role in solving equations and inequalities. In our previous article, we discussed how to find the complete factored form of the polynomial $-7m^6n^5 + 8n^3$. In this article, we will answer some frequently asked questions about factoring polynomials.

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

A: The greatest common factor (GCF) of two terms is the largest expression that divides both terms without leaving a remainder.

Q: How do I find the GCF of two terms?

A: To find the GCF of two terms, you can list the factors of each term and find the greatest common factor.

Q: What is the difference of squares formula?

A: The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you can rewrite the expression as $(a + b)(a - b)$.

Q: What is the complete factored form of a polynomial?

A: The complete factored form of a polynomial is the product of all its factors.

Q: How do I find the complete factored form of a polynomial?

A: To find the complete factored form of a polynomial, you can factor out the greatest common factor (GCF) of the two terms, and then factor the remaining expression using the difference of squares formula.

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) of the two terms
• Not applying the difference of squares formula correctly
• Not simplifying the expression after factoring

Q: How do I check if my factored form is correct?

A: To check if your factored form is correct, you can multiply the factors together and simplify the expression.

Q: What are some real-world applications of factoring polynomials?

A: Some real-world applications of factoring polynomials include:

• Solving equations and inequalities
• Finding the roots of a polynomial
• Factoring quadratic expressions

Q: Can you give an example of factoring a polynomial?

A: Yes, let's consider the polynomial $x^2 + 5x + 6$. We can factor this polynomial as $(x + 3)(x + 2)$.

Q: How do I factor a polynomial with a negative coefficient?

A: To factor a polynomial with a negative coefficient, you can factor out the negative sign and then factor the remaining expression.

Q: Can you give an example of factoring a polynomial with a negative coefficient?

A: Yes, let's consider the polynomial $-x^2 + 5x + 6$. We can factor this polynomial as $-(x^2 - 5x - 6)$, and then factor the remaining expression as $-(x - 3)(x + 2)$.

Conclusion

In this article, we have answered some frequently asked questions about factoring polynomials. We have discussed the greatest common factor (GCF), the difference of squares formula, and the complete factored form of a polynomial. We have also provided some examples of factoring polynomials and discussed some common mistakes to avoid.