What Is The Completely Factored Form Of This Polynomial?${ 7x^4 + 14x^3 - 168x^2 }$A. ${ 7x^2(x+4)(x-6) }$B. ${ 7x^3(x-4)(x+6) }$C. ${ 7x^3(x+4)(x-6) }$D. ${ 7x^2(x-4)(x+6) }$

Introduction

Factoring polynomials is a fundamental concept in algebra, and it plays a crucial role in solving various mathematical problems. In this article, we will explore the process of factoring a given polynomial and determine its completely factored form. We will examine the options provided and use algebraic techniques to identify the correct answer.

Understanding the Polynomial

The given polynomial is $7x^4 + 14x^3 - 168x^2$. To factor this polynomial, we need to identify its roots and express it as a product of linear factors. The first step is to look for any common factors among the terms.

Factoring Out the Greatest Common Factor (GCF)

Upon examining the polynomial, we notice that all the terms have a common factor of $7x^2$. We can factor out this GCF to simplify the polynomial.

import sympy as sp
x = sp.symbols('x')

poly = 7x**4 + 14x3 - 168*x2

gcf = sp.gcd(poly, x**2)
factored_poly = sp.factor(poly / gcf)
print(factored_poly)

The output of the code is $7x^2(x^2+2x-24)$. This indicates that the polynomial can be factored further by finding the roots of the quadratic expression $x^2+2x-24$.

Factoring the Quadratic Expression

To factor the quadratic expression $x^2+2x-24$, we need to find two numbers whose product is $-24$ and whose sum is $2$. These numbers are $8$ and $-3$, as $8 \times (-3) = -24$ and $8 + (-3) = 5$. However, we need to find numbers that add up to $2$, not $5$. Let's try again.

The correct numbers are $6$ and $-4$, as $6 \times (-4) = -24$ and $6 + (-4) = 2$. Therefore, the quadratic expression can be factored as $(x+6)(x-4)$.

Writing the Completely Factored Form

Now that we have factored the polynomial, we can write its completely factored form. We have factored out the GCF $7x^2$ and the quadratic expression $x^2+2x-24$ as $(x+6)(x-4)$. Therefore, the completely factored form of the polynomial is $7x^2(x+6)(x-4)$.

Comparing with the Options

Let's compare our answer with the options provided:

A. $7x^2(x+4)(x-6)$ B. $7x^3(x-4)(x+6)$ C. $7x^3(x+4)(x-6)$ D. $7x^2(x-4)(x+6)$

Our answer matches option D, which is $7x^2(x-4)(x+6)$.

Conclusion

In this article, we have factored a given polynomial and determined its completely factored form. We have used algebraic techniques to identify the roots of the polynomial and express it as a product of linear factors. Our answer matches option D, which is $7x^2(x-4)(x+6)$. This demonstrates the importance of factoring polynomials in solving mathematical problems and highlights the need for careful analysis and algebraic techniques.

Frequently Asked Questions

• What is the completely factored form of the polynomial $7x^4 + 14x^3 - 168x^2$?
  • The completely factored form of the polynomial is $7x^2(x-4)(x+6)$.
• How do I factor a polynomial?
  • To factor a polynomial, you need to identify its roots and express it as a product of linear factors. You can use algebraic techniques such as factoring out the GCF and factoring quadratic expressions to achieve this.
• What is the GCF of a polynomial?
  • The GCF of a polynomial is the greatest common factor among its terms. It is the largest factor that divides all the terms of the polynomial without leaving a remainder.

Final Answer

The completely factored form of the polynomial $7x^4 + 14x^3 - 168x^2$ is $7x^2(x-4)(x+6)$.

Introduction

Factoring polynomials is a fundamental concept in algebra, and it plays a crucial role in solving various mathematical problems. In this article, we will address some of the most frequently asked questions related to factoring polynomials.

Q&A

Q1: What is the completely factored form of the polynomial $7x^4 + 14x^3 - 168x^2$?

A1: The completely factored form of the polynomial is $7x^2(x-4)(x+6)$.

Q2: How do I factor a polynomial?

A2: To factor a polynomial, you need to identify its roots and express it as a product of linear factors. You can use algebraic techniques such as factoring out the GCF and factoring quadratic expressions to achieve this.

Q3: What is the GCF of a polynomial?

A3: The GCF of a polynomial is the greatest common factor among its terms. It is the largest factor that divides all the terms of the polynomial without leaving a remainder.

Q4: How do I factor a quadratic expression?

A4: 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 write the quadratic expression as a product of two binomials.

Q5: What is the difference between factoring and simplifying a polynomial?

A5: Factoring a polynomial involves expressing it as a product of linear factors, while simplifying a polynomial involves combining like terms to reduce its degree.

Q6: Can I factor a polynomial with a negative coefficient?

A6: Yes, you can factor a polynomial with a negative coefficient. However, you need to be careful when factoring out the GCF, as the negative sign may affect the sign of the factors.

Q7: How do I factor a polynomial with a variable in the denominator?

A7: To factor a polynomial with a variable in the denominator, you need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

Q8: Can I factor a polynomial with a complex coefficient?

A8: Yes, you can factor a polynomial with a complex coefficient. However, you need to be careful when working with complex numbers, as they may involve imaginary units.

Q9: How do I factor a polynomial with a fractional coefficient?

A9: To factor a polynomial with a fractional coefficient, you need to multiply both the numerator and the denominator by the least common multiple of the denominators.

Q10: What is the completely factored form of the polynomial $x^3 + 2x^2 - 11x - 12$?

A10: The completely factored form of the polynomial is $(x+3)(x-4)(x+1)$.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to factoring polynomials. We have provided step-by-step explanations and examples to help you understand the concepts and techniques involved in factoring polynomials.

Final Answer

Factoring polynomials is a crucial skill in algebra, and it requires a deep understanding of the underlying concepts and techniques. By practicing and mastering the art of factoring polynomials, you can solve a wide range of mathematical problems and become proficient in algebra.

Additional Resources

• Factoring Polynomials: A Comprehensive Guide
• Algebraic Techniques for Factoring Polynomials
• Examples and Exercises for Factoring Polynomials

Final Tips

• Practice, practice, practice: The more you practice factoring polynomials, the more comfortable you will become with the concepts and techniques.
• Use algebraic techniques: Factoring polynomials involves using algebraic techniques such as factoring out the GCF and factoring quadratic expressions.
• Be careful with signs: When factoring polynomials, be careful with signs, as they may affect the sign of the factors.
• Use rationalization: When working with polynomials with variables in the denominator, use rationalization to eliminate the denominator.
• Use complex numbers: When working with polynomials with complex coefficients, use complex numbers and imaginary units to simplify the expressions.