Find The Complete Factored Form Of The Polynomial: $-15a^3b - 40a^5b^4$Enter The Correct Answer.

Feb 26, 2025 by ADMIN 97 views

**Factoring Polynomials: A Comprehensive Guide**

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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 finding the complete factored form of the given polynomial: $-15a^3b - 40a^5b^4$ . We will break down the problem step by step and provide a clear explanation of the factoring process.

Understanding the Polynomial

The given polynomial is $-15a^3b - 40a^5b^4$ . To factor this polynomial, we need 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.

Identifying the Greatest Common Factor (GCF)

To find the GCF, we need to look for the common factors in both terms. In this case, both terms have a common factor of $-5a^3b$ . This is the greatest common factor of the two terms.

Factoring Out the GCF

Now that we have identified the GCF, we can factor it out of both terms. To do this, we divide each term by the GCF and multiply the result by the GCF.

import sympy as sp

# Define the variables
a, b = sp.symbols('a b')

# Define the polynomial
poly = -15*a**3*b - 40*a**5*b**4

# Factor out the GCF
gcf = -5*a**3*b
factored_poly = gcf * (3 + 8*a**2*b)

Simplifying the Factored Form

The factored form of the polynomial is $-5a^3b(3 + 8a^2b)$ . This is the complete factored form of the given polynomial.

Conclusion

In this article, we have demonstrated how to find the complete factored form of a polynomial. We identified the greatest common factor (GCF) of the two terms and factored it out of both terms. The resulting factored form is $-5a^3b(3 + 8a^2b)$ . This is a fundamental concept in algebra that is used extensively in mathematics and other fields.

Example Problems

Problem 1

Find the complete factored form of the polynomial: $6x^2y - 12x^3y^2$ .

Solution

The GCF of the two terms is $6x^2y$ . Factoring out the GCF, we get:

import sympy as sp

# Define the variables
x, y = sp.symbols('x y')

# Define the polynomial
poly = 6*x**2*y - 12*x**3*y**2

# Factor out the GCF
gcf = 6*x**2*y
factored_poly = gcf * (1 - 2*x*y)

The factored form of the polynomial is $6x^2y(1 - 2xy)$ .

Problem 2

Find the complete factored form of the polynomial: $9a^2b^2 - 16a^3b^3$ .

Solution

The GCF of the two terms is $a^2b^2$ . Factoring out the GCF, we get:

import sympy as sp

# Define the variables
a, b = sp.symbols('a b')

# Define the polynomial
poly = 9*a**2*b**2 - 16*a**3*b**3

# Factor out the GCF
gcf = a**2*b**2
factored_poly = gcf * (3 - 4*a*b)

The factored form of the polynomial is $a^2b^2(3 - 4ab)$ .

Tips and Tricks

When factoring a polynomial, always look for the greatest common factor (GCF) of the two terms.
Factor out the GCF by dividing each term by the GCF and multiplying the result by the GCF.
Simplify the factored form by combining like terms.

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have demonstrated how to find the complete factored form of a polynomial. We identified the greatest common factor (GCF) of the two terms and factored it out of both terms. The resulting factored form is $-5a^3b(3 + 8a^2b)$ . This is a fundamental concept in algebra that is used extensively in mathematics and other fields.

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Q&A: Factoring Polynomials

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This is a fundamental concept in algebra that is used extensively in mathematics and other fields.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to identify the greatest common factor (GCF) of the two terms. Once you have identified the GCF, you can factor it out of both terms by dividing each term by the GCF and multiplying the result by the GCF.

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

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

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

A: To find the GCF of two terms, you need to look for the common factors in both terms. The GCF is the product of the common factors.

Q: Can I factor a polynomial with more than two terms?

A: Yes, you can factor a polynomial with more than two terms. However, you need to identify the GCF of all the terms and factor it out of all the terms.

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 combining like terms to get a simpler 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 manually.

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 correctly
Factoring out the wrong term
Not simplifying the factored form
Not checking your work

Q: How do I check my work when factoring a polynomial?

A: To check your work when factoring a polynomial, you need to multiply the factored form by the GCF and simplify the result. If the result is the original polynomial, then your work is correct.

Q: Can I factor a polynomial with negative coefficients?

A: Yes, you can factor a polynomial with negative coefficients. However, you need to take into account the signs of the coefficients when factoring the polynomial.

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

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

Solving systems of equations
Finding the roots of a polynomial
Factoring quadratic equations
Solving optimization problems

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have provided a comprehensive guide to factoring polynomials, including a Q&A section that answers common questions and provides tips and tricks for factoring polynomials.

Example Problems

Problem 1

Factor the polynomial: $6x^2y - 12x^3y^2$ .

Solution

The GCF of the two terms is $6x^2y$ . Factoring out the GCF, we get:

import sympy as sp

# Define the variables
x, y = sp.symbols('x y')

# Define the polynomial
poly = 6*x**2*y - 12*x**3*y**2

# Factor out the GCF
gcf = 6*x**2*y
factored_poly = gcf * (1 - 2*x*y)

The factored form of the polynomial is $6x^2y(1 - 2xy)$ .

Problem 2

Factor the polynomial: $9a^2b^2 - 16a^3b^3$ .

Solution

The GCF of the two terms is $a^2b^2$ . Factoring out the GCF, we get:

import sympy as sp

# Define the variables
a, b = sp.symbols('a b')

# Define the polynomial
poly = 9*a**2*b**2 - 16*a**3*b**3

# Factor out the GCF
gcf = a**2*b**2
factored_poly = gcf * (3 - 4*a*b)

The factored form of the polynomial is $a^2b^2(3 - 4ab)$ .

Tips and Tricks

When factoring a polynomial, always look for the greatest common factor (GCF) of the two terms.
Factor out the GCF by dividing each term by the GCF and multiplying the result by the GCF.
Simplify the factored form by combining like terms.
Check your work by multiplying the factored form by the GCF and simplifying the result.