Which Is The Completely Factored Form Of $12x^3 - 60x^2 + 4x - 20$?A. 4 ( 3 X 2 − 1 ) ( X − 5 4(3x^2 - 1)(x - 5 4 ( 3 X 2 − 1 ) ( X − 5 ]B. 4 X ( 3 X 2 + 1 ) ( X − 5 4x(3x^2 + 1)(x - 5 4 X ( 3 X 2 + 1 ) ( X − 5 ]C. 4 X ( 3 X 2 − 1 ) ( X + 5 4x(3x^2 - 1)(x + 5 4 X ( 3 X 2 − 1 ) ( X + 5 ]D. 4 ( 3 X 2 + 1 ) ( X − 5 4(3x^2 + 1)(x - 5 4 ( 3 X 2 + 1 ) ( X − 5 ]

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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 explore the process of factoring polynomials and apply it to the given polynomial $12x^3 - 60x^2 + 4x - 20$. We will examine each option and determine which one is the completely factored form of the given polynomial.

What is Factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials. It involves finding the factors of the polynomial, which are the numbers or expressions that multiply together to give the original polynomial. Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Types of Factoring

There are several types of factoring, including:

  • Greatest Common Factor (GCF) Factoring: This involves factoring out the greatest common factor of the terms in the polynomial.
  • Difference of Squares Factoring: This involves factoring the difference of two squares, which is a polynomial of the form a2−b2a^2 - b^2.
  • Sum and Difference of Cubes Factoring: This involves factoring the sum or difference of two cubes, which is a polynomial of the form a3+b3a^3 + b^3 or a3−b3a^3 - b^3.
  • Grouping Factoring: This involves grouping the terms in the polynomial into pairs and factoring each pair.

Factoring the Given Polynomial

To factor the given polynomial $12x^3 - 60x^2 + 4x - 20$, we can start by looking for the greatest common factor of the terms. In this case, the greatest common factor is 4, so we can factor out 4 from each term:

12x3−60x2+4x−20=4(3x3−15x2+x−5)12x^3 - 60x^2 + 4x - 20 = 4(3x^3 - 15x^2 + x - 5)

Next, we can look for other factors of the polynomial. We can try to factor the polynomial by grouping the terms into pairs:

3x3−15x2+x−5=(3x3−15x2)+(x−5)3x^3 - 15x^2 + x - 5 = (3x^3 - 15x^2) + (x - 5)

We can factor out a common factor from each pair:

3x3−15x2=3x2(x−5)3x^3 - 15x^2 = 3x^2(x - 5)

x−5=x−5x - 5 = x - 5

Now, we can combine the two factors:

3x3−15x2+x−5=3x2(x−5)+(x−5)3x^3 - 15x^2 + x - 5 = 3x^2(x - 5) + (x - 5)

We can factor out a common factor from the two terms:

3x2(x−5)+(x−5)=(3x2+1)(x−5)3x^2(x - 5) + (x - 5) = (3x^2 + 1)(x - 5)

Therefore, the completely factored form of the given polynomial is:

12x3−60x2+4x−20=4(3x2+1)(x−5)12x^3 - 60x^2 + 4x - 20 = 4(3x^2 + 1)(x - 5)

Conclusion

In conclusion, factoring polynomials is a powerful tool in algebra that allows us to simplify complex expressions and solve equations. By applying the different types of factoring, we can factor the given polynomial $12x^3 - 60x^2 + 4x - 20$ into its completely factored form: 4(3x2+1)(x−5)4(3x^2 + 1)(x - 5). This result is consistent with option D, which is the correct answer.

Final Answer

The completely factored form of the polynomial $12x^3 - 60x^2 + 4x - 20$ is:

4(3x2+1)(x−5)4(3x^2 + 1)(x - 5)

This is the correct answer, which is option D.

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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 our previous article, we explored the process of factoring polynomials and applied it to the given polynomial $12x^3 - 60x^2 + 4x - 20$. In this article, we will answer some frequently asked questions about factoring polynomials.

Q&A

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

A: The greatest common factor (GCF) of a polynomial is the largest expression that divides each term of the polynomial without leaving a remainder.

Q: How do I factor a polynomial using the GCF method?

A: To factor a polynomial using the GCF method, you need to identify the greatest common factor of the terms in the polynomial. Once you have identified the GCF, you can factor it out from each term.

Q: What is the difference of squares formula?

A: The difference of squares formula is a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b).

Q: How do I factor a polynomial using the difference of squares formula?

A: To factor a polynomial using the difference of squares formula, you need to identify if the polynomial is in the form of a2−b2a^2 - b^2. If it is, you can apply the formula to factor the polynomial.

Q: What is the sum and difference of cubes formula?

A: The sum and difference of cubes formula is a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2).

Q: How do I factor a polynomial using the sum and difference of cubes formula?

A: To factor a polynomial using the sum and difference of cubes formula, you need to identify if the polynomial is in the form of a3+b3a^3 + b^3 or a3−b3a^3 - b^3. If it is, you can apply the formula to factor the polynomial.

Q: What is the grouping method of factoring?

A: The grouping method of factoring involves grouping the terms in the polynomial into pairs and factoring each pair.

Q: How do I factor a polynomial using the grouping method?

A: To factor a polynomial using the grouping method, you need to group the terms in the polynomial into pairs and factor each pair.

Common Mistakes to Avoid

When factoring polynomials, there are several common mistakes to avoid:

  • Not identifying the GCF: Failing to identify the greatest common factor of the terms in the polynomial can lead to incorrect factoring.
  • Not applying the correct formula: Applying the wrong formula or not applying the correct formula can lead to incorrect factoring.
  • Not checking the factors: Failing to check the factors of the polynomial can lead to incorrect factoring.

Conclusion

In conclusion, factoring polynomials is a powerful tool in algebra that allows us to simplify complex expressions and solve equations. By understanding the different types of factoring and avoiding common mistakes, you can become proficient in factoring polynomials.

Final Tips

  • Practice, practice, practice: The more you practice factoring polynomials, the more comfortable you will become with the different types of factoring.
  • Use online resources: There are many online resources available that can help you learn and practice factoring polynomials.
  • Seek help when needed: Don't be afraid to seek help when you are struggling with factoring polynomials.