Which Shows One Way To Determine The Factors Of $x 3-12x 2-2x+24$ By Grouping?A. X ( X 2 − 12 ) + 2 ( X 2 − 12 X(x^2-12) + 2(x^2-12 X ( X 2 − 12 ) + 2 ( X 2 − 12 ]B. X ( X 2 − 12 ) − 2 ( X 2 − 12 X(x^2-12) - 2(x^2-12 X ( X 2 − 12 ) − 2 ( X 2 − 12 ]C. X 2 ( X − 12 ) + 2 ( X − 12 X^2(x-12) + 2(x-12 X 2 ( X − 12 ) + 2 ( X − 12 ]D. X 2 ( X − 12 ) − 2 ( X − 12 X^2(x-12) - 2(x-12 X 2 ( X − 12 ) − 2 ( X − 12 ]

Introduction

In algebra, factoring polynomials is a crucial skill that helps us simplify complex expressions and solve equations. One method of factoring polynomials is by grouping, which involves rearranging the terms of a polynomial to facilitate factoring. In this article, we will explore how to determine the factors of a given polynomial by grouping.

Understanding the Concept of Grouping

Grouping is a factoring technique that involves rearranging the terms of a polynomial to create two or more groups of terms that can be factored separately. This method is particularly useful when the polynomial has multiple terms and no obvious common factors.

The Given Polynomial

The given polynomial is $x^3-12x2-2x+24$. Our goal is to determine the factors of this polynomial by grouping.

Step 1: Rearrange the Terms

To begin, we need to rearrange the terms of the polynomial to create two groups of terms. We can do this by grouping the first two terms and the last two terms separately.

$$x^3-12x^2-2x+24 = (x^3-12x^2) + (-2x+24)$$

Step 2: Factor Out Common Factors

Now that we have rearranged the terms, we can factor out common factors from each group. In the first group, we can factor out $x^2$, while in the second group, we can factor out -2.

$$(x^3-12x^2) + (-2x+24) = x^2(x-12) + 2(x-12)$$

Step 3: Factor Out the Common Binomial

Now that we have factored out common factors from each group, we can factor out the common binomial $(x-12)$ from both groups.

$$x^2(x-12) + 2(x-12) = (x^2+2)(x-12)$$

Conclusion

In conclusion, we have successfully determined the factors of the given polynomial by grouping. The factors are $(x^2+2)$ and $(x-12)$.

Comparing the Options

Now that we have determined the factors of the polynomial, let's compare our result with the given options.

A. $x(x^2-12) + 2(x^2-12)$ B. $x(x^2-12) - 2(x^2-12)$ C. $x^2(x-12) + 2(x-12)$ D. $x^2(x-12) - 2(x-12)$

Our result matches option C, which is $x^2(x-12) + 2(x-12)$.

Final Answer

The correct answer is C. $x^2(x-12) + 2(x-12)$.

Additional Tips and Examples

Here are some additional tips and examples to help you practice factoring polynomials by grouping.

• When rearranging the terms, try to create two groups of terms that have common factors.
• When factoring out common factors, make sure to factor out the greatest common factor (GCF) from each group.
• When factoring out the common binomial, make sure to factor out the binomial that is common to both groups.

Example 1

Factor the polynomial $x^3+8x^2-9x-72$ by grouping.

Solution

To factor the polynomial, we need to rearrange the terms to create two groups of terms.

$$x^3+8x^2-9x-72 = (x^3+8x^2) + (-9x-72)$$

Now, we can factor out common factors from each group.

$$(x^3+8x^2) + (-9x-72) = x^2(x+8) + (-9)(x+8)$$

Finally, we can factor out the common binomial $(x+8)$ from both groups.

$$x^2(x+8) + (-9)(x+8) = (x^2-9)(x+8)$$

Example 2

Factor the polynomial $x^3-16x^2-9x+144$ by grouping.

Solution

To factor the polynomial, we need to rearrange the terms to create two groups of terms.

$$x^3-16x^2-9x+144 = (x^3-16x^2) + (-9x+144)$$

Now, we can factor out common factors from each group.

$$(x^3-16x^2) + (-9x+144) = x^2(x-16) + 16(x-16)$$

Finally, we can factor out the common binomial $(x-16)$ from both groups.

$$x^2(x-16) + 16(x-16) = (x^2+16)(x-16)$$

Conclusion

Frequently Asked Questions

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor polynomials by rearranging the terms into two or more groups, and then factoring out common factors from each group.

Q: How do I determine the factors of a polynomial by grouping?

A: To determine the factors of a polynomial by grouping, follow these steps:

1. Rearrange the terms of the polynomial to create two or more groups of terms.
2. Factor out common factors from each group.
3. Factor out the common binomial from both groups.

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

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

• Not rearranging the terms correctly to create groups of terms with common factors.
• Not factoring out the greatest common factor (GCF) from each group.
• Not factoring out the common binomial from both groups.

Q: Can I use factoring by grouping to factor polynomials with multiple variables?

A: Yes, you can use factoring by grouping to factor polynomials with multiple variables. However, you may need to use additional techniques, such as factoring out the GCF or using the distributive property, to simplify the polynomial.

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

A: A polynomial can be factored by grouping if it has multiple terms and can be rearranged into two or more groups of terms with common factors.

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 additional techniques, such as factoring out the GCF or using the distributive property, to simplify the polynomial.

Q: How do I check if my factored form is correct?

A: To check if your factored form is correct, multiply the factors together and simplify the expression. If the result is the original polynomial, then your factored form is correct.

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

A: Yes, you can use factoring by grouping to factor polynomials with rational coefficients. However, you may need to use additional techniques, such as factoring out the GCF or using the distributive property, to simplify the polynomial.

Q: How do I apply factoring by grouping to solve equations?

A: To apply factoring by grouping to solve equations, follow these steps:

1. Factor the polynomial using factoring by grouping.
2. Set the factored form equal to zero and solve for the variable.
3. Check your solutions by substituting them back into the original equation.

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

A: Yes, you can use factoring by grouping to factor polynomials with complex coefficients. However, you may need to use additional techniques, such as factoring out the GCF or using the distributive property, to simplify the polynomial.

Conclusion

In conclusion, factoring by grouping is a powerful technique that can help us simplify complex expressions and solve equations. By understanding the steps involved in factoring by grouping, avoiding common mistakes, and applying this technique to solve equations, you can become proficient in factoring polynomials by grouping.

Additional Resources

For more information on factoring by grouping, check out the following resources:

• Khan Academy: Factoring by Grouping
• Mathway: Factoring by Grouping
• Wolfram Alpha: Factoring by Grouping

Practice Problems

Try these practice problems to test your skills in factoring by grouping:

• Factor the polynomial $x^3+8x^2-9x-72$ by grouping.
• Factor the polynomial $x^3-16x^2-9x+144$ by grouping.
• Factor the polynomial $x^3+27x^2-18x-216$ by grouping.

Answer Key

• $x^2(x+8) + (-9)(x+8) = (x^2-9)(x+8)$
• $x^2(x-16) + 16(x-16) = (x^2+16)(x-16)$
• $x^2(x+27) + (-18)(x+27) = (x^2-18)(x+27)$