Which Is The Completely Factored Form Of Her Polynomial?A. ${ 8x^2(x+3)^2\$} B. ${ 2x(4x^2+5)\$} C. ${ 2(4x^2+5)(x+3)\$} D. ${ 2(4x^2+5)(x+3)^2\$}

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Understanding Polynomial Factoring

Polynomial factoring is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and simplifying expressions. In this article, we will delve into the world of polynomial factoring and explore the completely factored form of a given polynomial.

What is a Completely Factored Form?

A completely factored form of a polynomial is an expression where the polynomial is written as a product of prime factors, also known as irreducible factors. These prime factors are polynomials that cannot be factored further into simpler polynomials. The completely factored form is unique for each polynomial and provides valuable insights into the polynomial's properties.

The Given Polynomial

The given polynomial is:

8x2(x+3)28x^2(x+3)^2

Analyzing the Polynomial

To determine the completely factored form of the polynomial, we need to analyze its structure. The polynomial consists of two terms: 8x28x^2 and (x+3)2(x+3)^2. The first term is a quadratic expression, while the second term is a squared binomial.

Factoring the Quadratic Expression

The quadratic expression 8x28x^2 can be factored as:

8x2=23⋅x28x^2 = 2^3 \cdot x^2

However, this is not a complete factorization, as we can further factor the expression by taking out a common factor of 2x2x:

8x2=2x⋅4x8x^2 = 2x \cdot 4x

Factoring the Squared Binomial

The squared binomial (x+3)2(x+3)^2 can be factored as:

(x+3)2=(x+3)(x+3)(x+3)^2 = (x+3)(x+3)

Combining the Factors

Now that we have factored the quadratic expression and the squared binomial, we can combine the factors to obtain the completely factored form of the polynomial:

8x2(x+3)2=2x⋅4x⋅(x+3)(x+3)8x^2(x+3)^2 = 2x \cdot 4x \cdot (x+3)(x+3)

Simplifying the Expression

We can simplify the expression by combining like terms:

2x⋅4x⋅(x+3)(x+3)=2x⋅4x⋅(x+3)22x \cdot 4x \cdot (x+3)(x+3) = 2x \cdot 4x \cdot (x+3)^2

Conclusion

In conclusion, the completely factored form of the given polynomial is:

2x⋅4x⋅(x+3)2=2(4x2+5)(x+3)22x \cdot 4x \cdot (x+3)^2 = 2(4x^2+5)(x+3)^2

This expression represents the polynomial in its simplest form, where the factors are irreducible and cannot be factored further.

Comparison with the Options

Let's compare our result with the given options:

  • A. 8x2(x+3)28x^2(x+3)^2
  • B. 2x(4x2+5)2x(4x^2+5)
  • C. 2(4x2+5)(x+3)2(4x^2+5)(x+3)
  • D. 2(4x2+5)(x+3)22(4x^2+5)(x+3)^2

Our result matches option D, which is:

2(4x2+5)(x+3)22(4x^2+5)(x+3)^2

This confirms that the completely factored form of the given polynomial is indeed option D.

Final Thoughts

In this article, we explored the concept of polynomial factoring and determined the completely factored form of a given polynomial. We analyzed the polynomial's structure, factored the quadratic expression and the squared binomial, and combined the factors to obtain the final expression. Our result matched option D, confirming that it is the correct completely factored form of the polynomial.

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Understanding Polynomial Factoring

Polynomial factoring is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and simplifying expressions. In this article, we will delve into the world of polynomial factoring and provide a comprehensive Q&A guide to help you understand the concept better.

Q&A: Polynomial Factoring

Q1: What is polynomial factoring?

A: Polynomial factoring is the process of expressing a polynomial as a product of simpler polynomials, called factors.

Q2: Why is polynomial factoring important?

A: Polynomial factoring is essential in solving equations, finding roots, and simplifying expressions. It helps us to identify the factors of a polynomial and understand its properties.

Q3: What are the different types of polynomial factoring?

A: There are several types of polynomial factoring, including:

  • Factoring out a greatest common factor (GCF)
  • Factoring by grouping
  • Factoring quadratics
  • Factoring polynomials with rational exponents

Q4: How do I factor a polynomial?

A: To factor a polynomial, you need to identify the factors of the polynomial and express it as a product of simpler polynomials. You can use various factoring techniques, such as factoring out a GCF, factoring by grouping, or factoring quadratics.

Q5: 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 an expression.

Q6: Can all polynomials be factored?

A: No, not all polynomials can be factored. Some polynomials are irreducible, meaning they cannot be factored into simpler polynomials.

Q7: How do I determine if a polynomial is irreducible?

A: To determine if a polynomial is irreducible, you need to check if it has any rational roots. If it does not have any rational roots, then it is likely to be irreducible.

Q8: What is the completely factored form of a polynomial?

A: The completely factored form of a polynomial is an expression where the polynomial is written as a product of prime factors, also known as irreducible factors.

Q9: How do I find the completely factored form of a polynomial?

A: To find the completely factored form of a polynomial, you need to factor the polynomial using various factoring techniques and express it as a product of prime factors.

Q10: What are some common mistakes to avoid when factoring polynomials?

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

  • Not identifying the factors of the polynomial
  • Not using the correct factoring technique
  • Not checking for rational roots
  • Not expressing the polynomial in its simplest form

Conclusion

In conclusion, polynomial factoring is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and simplifying expressions. By understanding the different types of polynomial factoring and avoiding common mistakes, you can become proficient in factoring polynomials and solve complex algebraic problems.

Final Thoughts

Polynomial factoring is a powerful tool in algebra that can help you solve complex problems and simplify expressions. By mastering the art of polynomial factoring, you can become a proficient algebraist and tackle even the most challenging problems with confidence.