Factor { -8x^3 - 2x^2 - 12x - 3$}$ By Grouping. What Is The Resulting Expression?A. { (2x^2 - 3)(4x + 1)$}$B. { (-2x^2 - 3)(-4x + 1)$}$C. { (2x^2 - 3)(-4x + 1)$}$D. { (-2x^2 - 3)(4x + 1)$}$

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Introduction

Factoring polynomials is an essential skill in algebra, and one of the techniques used to factor polynomials is grouping. Grouping involves factoring a polynomial by grouping terms that have common factors. In this article, we will learn how to factor the polynomial −8x3−2x2−12x−3{-8x^3 - 2x^2 - 12x - 3} by grouping.

Understanding the Polynomial

Before we start factoring, let's take a closer look at the polynomial −8x3−2x2−12x−3{-8x^3 - 2x^2 - 12x - 3}. This polynomial has four terms, and each term has a negative sign. To factor this polynomial, we need to find two terms that have a common factor.

Step 1: Grouping the Terms

To factor the polynomial by grouping, we need to group the terms in pairs. Let's group the first two terms and the last two terms:

−8x3−2x2−12x−3=(−8x3−2x2)−(12x+3){-8x^3 - 2x^2 - 12x - 3 = (-8x^3 - 2x^2) - (12x + 3)}

Step 2: Factoring Out Common Factors

Now that we have grouped the terms, let's factor out common factors from each group. From the first group, we can factor out −2x2{-2x^2}, and from the second group, we can factor out −3{-3}:

−8x3−2x2−12x−3=−2x2(4x1+1)−3(4x1+1){-8x^3 - 2x^2 - 12x - 3 = -2x^2(4x^1 + 1) - 3(4x^1 + 1)}

Step 3: Factoring Out the Common Binomial Factor

Now that we have factored out common factors from each group, we can see that both groups have a common binomial factor, (4x+1){(4x + 1)}. We can factor this binomial factor out of both groups:

−8x3−2x2−12x−3=−(2x2+3)(4x+1){-8x^3 - 2x^2 - 12x - 3 = -(2x^2 + 3)(4x + 1)}

Step 4: Simplifying the Expression

The final step is to simplify the expression by removing any negative signs that are not necessary. In this case, we can remove the negative sign from the first term:

−8x3−2x2−12x−3=(2x2+3)(−4x+1){-8x^3 - 2x^2 - 12x - 3 = (2x^2 + 3)(-4x + 1)}

Conclusion

In this article, we learned how to factor the polynomial −8x3−2x2−12x−3{-8x^3 - 2x^2 - 12x - 3} by grouping. We grouped the terms in pairs, factored out common factors, and factored out a common binomial factor. The resulting expression is (2x2+3)(−4x+1){(2x^2 + 3)(-4x + 1)}.

Answer

The correct answer is:

{(2x^2 + 3)(-4x + 1)$]

This answer matches option C.

Discussion

Factoring polynomials by grouping is a powerful technique that can be used to factor polynomials with multiple terms. By grouping terms that have common factors, we can factor out common factors and simplify the expression. In this article, we learned how to factor the polynomial [-8x^3 - 2x^2 - 12x - 3}$ by grouping, and we saw that the resulting expression is (2x2+3)(−4x+1){(2x^2 + 3)(-4x + 1)}.

Common Mistakes

When factoring polynomials by grouping, there are several common mistakes that students make. One common mistake is to forget to factor out common factors from each group. Another common mistake is to forget to factor out the common binomial factor. To avoid these mistakes, it's essential to carefully read the polynomial and identify the common factors.

Tips and Tricks

When factoring polynomials by grouping, here are some tips and tricks to keep in mind:

  • Always group the terms in pairs.
  • Factor out common factors from each group.
  • Factor out the common binomial factor.
  • Simplify the expression by removing any unnecessary negative signs.

By following these tips and tricks, you can master the technique of factoring polynomials by grouping and simplify complex expressions with ease.

Real-World Applications

Factoring polynomials by grouping has several real-world applications. For example, in physics, factoring polynomials is used to solve problems involving motion and energy. In engineering, factoring polynomials is used to design and optimize systems. In economics, factoring polynomials is used to model and analyze economic systems.

Conclusion

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor polynomials by grouping terms that have common factors. This technique involves factoring out common factors from each group and then factoring out a common binomial factor.

Q: How do I factor a polynomial by grouping?

A: To factor a polynomial by grouping, follow these steps:

  1. Group the terms in pairs.
  2. Factor out common factors from each group.
  3. Factor out the common binomial factor.
  4. Simplify the expression by removing any unnecessary negative signs.

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

A: Some common mistakes to avoid when factoring by grouping include:

  • Forgetting to factor out common factors from each group.
  • Forgetting to factor out the common binomial factor.
  • Not simplifying the expression by removing any unnecessary negative signs.

Q: What are some tips and tricks for factoring by grouping?

A: Some tips and tricks for factoring by grouping include:

  • Always group the terms in pairs.
  • Factor out common factors from each group.
  • Factor out the common binomial factor.
  • Simplify the expression by removing any unnecessary negative signs.

Q: When should I use factoring by grouping?

A: You should use factoring by grouping when you have a polynomial with multiple terms and you want to factor it. This technique is particularly useful when the polynomial has a common binomial factor.

Q: Can I use factoring by grouping with polynomials that have negative coefficients?

A: Yes, you can use factoring by grouping with polynomials that have negative coefficients. However, you may need to factor out a negative sign from one of the groups.

Q: How do I know if a polynomial can be factored by grouping?

A: You can determine if a polynomial can be factored by grouping by looking for common factors among the terms. If you can find common factors, you can factor the polynomial by grouping.

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

A: Some real-world applications of factoring by grouping include:

  • Solving problems involving motion and energy in physics.
  • Designing and optimizing systems in engineering.
  • Modeling and analyzing economic systems in economics.

Q: Can I use factoring by grouping with polynomials that have fractional coefficients?

A: Yes, you can use factoring by grouping with polynomials that have fractional coefficients. However, you may need to factor out a common factor from each group.

Q: How do I simplify an expression that has been factored by grouping?

A: To simplify an expression that has been factored by grouping, remove any unnecessary negative signs and combine like terms.

Q: What are some common polynomials that can be factored by grouping?

A: Some common polynomials that can be factored by grouping include:

  • x2+5x+6{x^2 + 5x + 6}
  • x2−7x+12{x^2 - 7x + 12}
  • x2+2x−15{x^2 + 2x - 15}

Q: Can I use factoring by grouping with polynomials that have imaginary coefficients?

A: No, you cannot use factoring by grouping with polynomials that have imaginary coefficients. Factoring by grouping only works with polynomials that have real coefficients.

Q: How do I know if a polynomial has been factored correctly by grouping?

A: To determine if a polynomial has been factored correctly by grouping, multiply the factors together and simplify the expression. If the result is the original polynomial, then the factoring was done correctly.