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 out common factors from two or more terms in a polynomial expression. 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 we can see that the first and third terms have a common factor of −4x-4x, while the second and fourth terms have a common factor of −1-1.

Step 1: Factor Out Common Factors

To factor the polynomial by grouping, we need to factor out common factors from two or more terms. In this case, we can factor out −4x-4x from the first and third terms, and −1-1 from the second and fourth terms.

-8x^3 - 2x^2 - 12x - 3
= (-4x)(2x^2) - (-1)(2x^2) - (-4x)(3) - (-1)(3)

Step 2: Factor Out Common Binomials

Now that we have factored out common factors, we can factor out common binomials from the two groups of terms. In this case, we can factor out (2x2−3)(2x^2 - 3) from the first group, and (−4x+1)(-4x + 1) from the second group.

(-4x)(2x^2) - (-1)(2x^2) - (-4x)(3) - (-1)(3)
= (2x^2 - 3)(-4x) + (2x^2 - 3)(1) + (4x - 1)(3) + (4x - 1)(1)

Step 3: Simplify the Expression

Now that we have factored out common binomials, we can simplify the expression by combining like terms.

(2x^2 - 3)(-4x) + (2x^2 - 3)(1) + (4x - 1)(3) + (4x - 1)(1)
= (2x^2 - 3)(-4x + 1) + (4x - 1)(3 + 1)
= (2x^2 - 3)(-4x + 1) + (4x - 1)(4)

Conclusion

In conclusion, we have factored the polynomial −8x3−2x2−12x−3-8x^3 - 2x^2 - 12x - 3 by grouping. The resulting expression is (2x2−3)(−4x+1)+(4x−1)(4)(2x^2 - 3)(-4x + 1) + (4x - 1)(4). This expression can be further simplified by combining like terms.

Answer

The correct answer is C. (2x2−3)(−4x+1)(2x^2 - 3)(-4x + 1).

Discussion

This problem requires the student to factor a polynomial by grouping, which is an essential skill in algebra. The student needs to identify the common factors and binomials, and then simplify the expression by combining like terms. This problem also requires the student to have a good understanding of algebraic expressions and equations.

Tips and Tricks

  • When factoring by grouping, it's essential to identify the common factors and binomials.
  • Use the distributive property to expand the expression and simplify it.
  • Combine like terms to simplify the expression.

Practice Problems

  • Factor the polynomial 6x3+2x2+12x+46x^3 + 2x^2 + 12x + 4 by grouping.
  • Factor the polynomial −9x3−3x2−27x−9-9x^3 - 3x^2 - 27x - 9 by grouping.

Conclusion

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor polynomials by grouping two or more terms together and factoring out common factors from each group.

Q: How do I identify the common factors and binomials in a polynomial?

A: To identify the common factors and binomials, look for terms that have a common factor or binomial. You can use the distributive property to expand the expression and simplify it.

Q: What is the distributive property?

A: The distributive property is a rule that states that a single term can be distributed to multiple terms. For example, a(b+c)=ab+aca(b + c) = ab + ac.

Q: How do I simplify the expression after factoring by grouping?

A: To simplify the expression, combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x22x^2 and 3x23x^2 are like terms because they both have the variable xx and the exponent 22.

Q: Can I factor a polynomial by grouping if it has more than four terms?

A: Yes, you can factor a polynomial by grouping if it has more than four terms. However, it may be more difficult to identify the common factors and binomials.

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

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

  • Not identifying the common factors and binomials
  • Not using the distributive property to expand the expression
  • Not combining like terms to simplify the expression

Q: How do I know if I have factored a polynomial correctly?

A: To check if you have factored a polynomial correctly, multiply the factors together and simplify the expression. If the result is the original polynomial, then you have factored it correctly.

Q: Can I use factoring by grouping to factor polynomials with negative coefficients?

A: Yes, you can use factoring by grouping to factor polynomials with negative coefficients. However, you may need to use the distributive property to expand the expression and simplify it.

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

A: Factoring by grouping has many real-world applications, including:

  • Solving systems of equations
  • Finding the roots of a polynomial
  • Factoring quadratic expressions

Q: Can I use factoring by grouping to factor polynomials with fractional coefficients?

A: Yes, you can use factoring by grouping to factor polynomials with fractional coefficients. However, you may need to use the distributive property to expand the expression and simplify it.

Conclusion

In this article, we have answered some common questions about factoring by grouping. We have discussed how to identify common factors and binomials, how to simplify the expression, and how to avoid common mistakes. We have also discussed some real-world applications of factoring by grouping.