What Are The Factors Of $x^3+125$? If The Polynomial Cannot Be Factored, Select Prime.A) $(x+5)\left(x^2-5x+25\right)$ B) \$(x-5)\left(x^2+5x+25\right)$[/tex\] C) $(x+5)^3$ D) Prime

Feb 27, 2025 by ADMIN 192 views

Introduction

Factoring polynomials is an essential skill in algebra, and it can be a challenging task, especially when dealing with cubic expressions. In this article, we will explore the factors of the polynomial $x^3+125$ and determine whether it can be factored or not.

Understanding the Polynomial

The given polynomial is $x^3+125$. To factor this expression, we need to look for common factors and use various factoring techniques. Let's start by examining the structure of the polynomial.

Factoring the Polynomial

We can rewrite the polynomial as $x^3+125 = (x+5)(x^2-5x+25)$ using the sum of cubes formula. This formula states that $a^3+b3 = (a+b)(a^2-ab+b2)$.

Verifying the Factorization

To verify the factorization, we can multiply the two factors together:

(x+5)(x^2-5x+25) = x(x^2-5x+25) + 5(x^2-5x+25)

Expanding the expression, we get:

x^3-5x^2+25x+5x^2-25x+125

Simplifying the expression, we get:

x^3+125

This confirms that the factorization is correct.

Conclusion

Based on the factorization, we can conclude that the factors of $x^3+125$ are $(x+5)(x^2-5x+25)$. This means that the polynomial can be factored, and the correct answer is option A.

Comparison with Other Options

Let's compare the correct factorization with the other options:

Option B: $(x-5)\left(x^2+5x+25\right)$ is incorrect because it does not match the correct factorization.
Option C: $(x+5)^3$ is incorrect because it is not a factorization of the polynomial.
Option D: Prime is incorrect because the polynomial can be factored.

Final Answer

The final answer is option A: $(x+5)\left(x^2-5x+25\right)$.

Additional Tips and Tricks

When factoring polynomials, it's essential to look for common factors and use various factoring techniques. Here are some additional tips and tricks:

Look for common factors: Check if there are any common factors in the polynomial, such as a greatest common factor (GCF).
Use the sum of cubes formula: If the polynomial is in the form $a^3+b3$, use the sum of cubes formula to factor it.
Use the difference of cubes formula: If the polynomial is in the form $a^3-b3$, use the difference of cubes formula to factor it.
Use synthetic division: If the polynomial is in the form $ax^2+bx+c$, use synthetic division to factor it.

By following these tips and tricks, you can become proficient in factoring polynomials and solve complex algebraic expressions.

Conclusion

In conclusion, the factors of $x^3+125$ are $(x+5)(x^2-5x+25)$. This means that the polynomial can be factored, and the correct answer is option A. By following the tips and tricks outlined in this article, you can become proficient in factoring polynomials and solve complex algebraic expressions.

Frequently Asked Questions

Q: What is the sum of cubes formula? A: The sum of cubes formula is $a^3+b3 = (a+b)(a^2-ab+b2)$.

Q: What is the difference of cubes formula? A: The difference of cubes formula is $a^3-b3 = (a-b)(a^2+ab+b2)$.

Q: How do I factor a polynomial using synthetic division? A: To factor a polynomial using synthetic division, follow these steps:

Write the polynomial in the form $ax^2+bx+c$.
Choose a value for the variable, such as $x=1$.
Divide the polynomial by the chosen value using synthetic division.
Write the result as a quotient and a remainder.
Factor the quotient and the remainder.

By following these steps, you can factor a polynomial using synthetic division.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman

Note: The references provided are for general information and are not specific to the topic of factoring polynomials.

Final Thoughts

Factoring polynomials is an essential skill in algebra, and it can be a challenging task, especially when dealing with cubic expressions. By following the tips and tricks outlined in this article, you can become proficient in factoring polynomials and solve complex algebraic expressions. Remember to look for common factors, use the sum of cubes formula, and use synthetic division to factor polynomials. With practice and patience, you can master the art of factoring polynomials.

Introduction

Factoring polynomials is an essential skill in algebra, and it can be a challenging task, especially when dealing with cubic expressions. In this article, we will answer some frequently asked questions about factoring polynomials and provide additional tips and tricks to help you master this skill.

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 reduce its degree.

Q: How do I determine if a polynomial can be factored?

A: To determine if a polynomial can be factored, look for common factors, such as a greatest common factor (GCF), and use various factoring techniques, such as the sum of cubes formula and synthetic division.

Q: What is the sum of cubes formula?

A: The sum of cubes formula is $a^3+b3 = (a+b)(a^2-ab+b2)$.

Q: What is the difference of cubes formula?

A: The difference of cubes formula is $a^3-b3 = (a-b)(a^2+ab+b2)$.

Q: How do I factor a polynomial using synthetic division?

A: To factor a polynomial using synthetic division, follow these steps:

Write the polynomial in the form $ax^2+bx+c$.
Choose a value for the variable, such as $x=1$.
Divide the polynomial by the chosen value using synthetic division.
Write the result as a quotient and a remainder.
Factor the quotient and the remainder.

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

A: The greatest common factor (GCF) of a polynomial is the largest polynomial that divides each term of the polynomial.

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

A: To find the GCF of a polynomial, look for common factors among the terms and multiply them together.

Q: What is the difference between a monomial and a polynomial?

A: A monomial is a single term, such as $x^2$, while a polynomial is a sum of monomials, such as $x^2+3x-4$.

Q: How do I factor a polynomial with multiple variables?

A: To factor a polynomial with multiple variables, use the distributive property to expand the polynomial and then factor out common factors.

Q: What is the relationship between factoring and graphing a polynomial?

A: Factoring a polynomial can help you graph it by identifying its roots and using them to plot the graph.

Q: How do I use factoring to solve a system of equations?

A: To use factoring to solve a system of equations, factor the equations and then solve for the variables.

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

A: Some common mistakes to avoid when factoring polynomials include:

Not looking for common factors
Not using the correct factoring technique
Not checking the result for errors

Q: How can I practice factoring polynomials?

A: You can practice factoring polynomials by working through examples and exercises in a textbook or online resource, or by creating your own problems to solve.

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

A: Factoring polynomials has many real-world applications, including:

Solving systems of equations
Graphing functions
Finding roots of polynomials
Factoring quadratic expressions

Conclusion

Additional Resources

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman
[4] Khan Academy: Factoring Polynomials
[5] Mathway: Factoring Polynomials

Note: The resources provided are for general information and are not specific to the topic of factoring polynomials.