Factor The Polynomial 12 X 2 − 6 X 12x^2 - 6x 12 X 2 − 6 X Completely.A. 2 ( 6 X − 3 ) 2 2(6x - 3)^2 2 ( 6 X − 3 ) 2 B. X ( 12 X − 6 X(12x - 6 X ( 12 X − 6 ] C. 6 X ( 2 X − 1 6x(2x - 1 6 X ( 2 X − 1 ] D. 3 X ( 12 X 2 − 2 3x(12x^2 - 2 3 X ( 12 X 2 − 2 ] E. 12 X ( X − 1 12x(x - 1 12 X ( X − 1 ]

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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 factoring the polynomial 12x2−6x12x^2 - 6x completely. We will explore the different methods of factoring and provide step-by-step solutions to help you understand the process.

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 skill in algebra, as it allows us to simplify complex expressions and solve equations.

The Polynomial 12x2−6x12x^2 - 6x

The given polynomial is 12x2−6x12x^2 - 6x. To factor this polynomial, we need to find the greatest common factor (GCF) of the two terms. The GCF is the largest number or expression that divides both terms evenly.

Step 1: Find the Greatest Common Factor (GCF)

To find the GCF, we need to identify the common factors of the two terms. In this case, the common factor is 6. Therefore, the GCF is 6.

Step 2: Factor Out the GCF

Now that we have found the GCF, we can factor it out of the polynomial. To do this, we divide each term by the GCF and write the result as a product of the GCF and the remaining expression.

12x^2 - 6x = 6(2x^2 - x)

Step 3: Factor the Remaining Expression

The remaining expression is 2x2−x2x^2 - x. To factor this expression, we need to find two numbers whose product is 2 and whose sum is -1. These numbers are -2 and 1.

2x^2 - x = 2x(x - 1/2)

Step 4: Write the Final Factored Form

Now that we have factored the remaining expression, we can write the final factored form of the polynomial.

12x^2 - 6x = 6(2x(x - 1/2))

Simplifying the Factored Form

We can simplify the factored form by multiplying the numbers inside the parentheses.

12x^2 - 6x = 6(2x^2 - x)

Conclusion

In this article, we have factored the polynomial 12x2−6x12x^2 - 6x completely. We have found the greatest common factor (GCF) of the two terms, factored it out, and then factored the remaining expression. The final factored form of the polynomial is 6(2x2−x)6(2x^2 - x).

Answer

The correct answer is B. x(12x−6)x(12x - 6).

Why is this the correct answer?

The correct answer is B. x(12x−6)x(12x - 6) because it is the only option that matches the final factored form of the polynomial. The other options do not match the final factored form, and therefore, are incorrect.

Tips and Tricks

Here are some tips and tricks to help you factor polynomials:

  • Always look for the greatest common factor (GCF) of the two terms.
  • Factor out the GCF and then factor the remaining expression.
  • Use the distributive property to simplify the factored form.
  • Check your work by multiplying the factored form back to the original polynomial.

Common Mistakes

Here are some common mistakes to avoid when factoring polynomials:

  • Not finding the greatest common factor (GCF) of the two terms.
  • Not factoring out the GCF.
  • Not factoring the remaining expression.
  • Not simplifying the factored form.

Conclusion

Q&A: Factoring the Polynomial 12x2−6x12x^2 - 6x

Q: What is the greatest common factor (GCF) of the two terms in the polynomial 12x2−6x12x^2 - 6x?

A: The greatest common factor (GCF) of the two terms is 6.

Q: How do you factor out the GCF from the polynomial 12x2−6x12x^2 - 6x?

A: To factor out the GCF, we divide each term by the GCF and write the result as a product of the GCF and the remaining expression.

12x^2 - 6x = 6(2x^2 - x)

Q: What is the remaining expression after factoring out the GCF?

A: The remaining expression is 2x2−x2x^2 - x.

Q: How do you factor the remaining expression 2x2−x2x^2 - x?

A: To factor the remaining expression, we need to find two numbers whose product is 2 and whose sum is -1. These numbers are -2 and 1.

2x^2 - x = 2x(x - 1/2)

Q: What is the final factored form of the polynomial 12x2−6x12x^2 - 6x?

A: The final factored form of the polynomial is 6(2x2−x)6(2x^2 - x).

Q: Why is option B the correct answer?

A: Option B is the correct answer because it matches the final factored form of the polynomial.

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

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

  • Not finding the greatest common factor (GCF) of the two terms.
  • Not factoring out the GCF.
  • Not factoring the remaining expression.
  • Not simplifying the factored form.

Q: How can I simplify the factored form of the polynomial?

A: You can simplify the factored form by multiplying the numbers inside the parentheses.

12x^2 - 6x = 6(2x^2 - x)

Q: What are some tips and tricks to help me factor polynomials?

A: Some tips and tricks to help you factor polynomials include:

  • Always look for the greatest common factor (GCF) of the two terms.
  • Factor out the GCF and then factor the remaining expression.
  • Use the distributive property to simplify the factored form.
  • Check your work by multiplying the factored form back to the original polynomial.

Conclusion

Factoring polynomials is an essential skill in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have factored the polynomial 12x2−6x12x^2 - 6x completely. We have found the greatest common factor (GCF) of the two terms, factored it out, and then factored the remaining expression. The final factored form of the polynomial is 6(2x2−x)6(2x^2 - x).