Factor $x^3 - 7x^2 - 5x + 35$ By Grouping. What Is The Resulting Expression?A. $\left(x^2 - 7\right)(x - 5)$B. \$\left(x^2 - 7\right)(x + 5)$[/tex\]C. $\left(x^2 - 5\right)(x - 7)$D. $\left(x^2 +

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Introduction

Factorizing a cubic expression can be a challenging task, but it becomes more manageable when we use the method of grouping. This method involves grouping the terms of the expression in a way that allows us to factor out common factors. In this article, we will learn how to factor the cubic expression $x^3 - 7x^2 - 5x + 35$ by grouping.

Understanding the Method of Grouping

The method of grouping involves dividing the terms of the expression into two groups, such that each group has two terms. We then factor out common factors from each group. The key to this method is to identify the common factors that can be factored out from each group.

Step 1: Divide the Terms into Two Groups

To factor the expression $x^3 - 7x^2 - 5x + 35$ by grouping, we first divide the terms into two groups:

(x3βˆ’7x2)βˆ’(5xβˆ’35)\left(x^3 - 7x^2\right) - \left(5x - 35\right)

Step 2: Factor Out Common Factors from Each Group

Now, we factor out common factors from each group. From the first group, we can factor out $x^2$:

x2(xβˆ’7)x^2\left(x - 7\right)

From the second group, we can factor out $-5$:

βˆ’5(xβˆ’7)-5\left(x - 7\right)

Step 3: Combine the Factored Groups

Now, we combine the factored groups:

x2(xβˆ’7)βˆ’5(xβˆ’7)x^2\left(x - 7\right) - 5\left(x - 7\right)

Step 4: Factor Out the Common Binomial Factor

We can see that both groups have a common binomial factor $(x - 7)$. We can factor this out:

(xβˆ’7)(x2βˆ’5)(x - 7)\left(x^2 - 5\right)

Conclusion

In this article, we learned how to factor the cubic expression $x^3 - 7x^2 - 5x + 35$ by grouping. We divided the terms into two groups, factored out common factors from each group, and then combined the factored groups. Finally, we factored out the common binomial factor to get the resulting expression.

The Resulting Expression

The resulting expression is:

(xβˆ’7)(x2βˆ’5)(x - 7)\left(x^2 - 5\right)

This is the correct answer among the given options.

Comparison with Other Options

Let's compare our resulting expression with the other options:

  • Option A: $\left(x^2 - 7\right)(x - 5)$
  • Option B: $\left(x^2 - 7\right)(x + 5)$
  • Option C: $\left(x^2 - 5\right)(x - 7)$
  • Option D: $\left(x^2 + 5\right)(x - 7)$

Our resulting expression matches with Option C: $\left(x^2 - 5\right)(x - 7)$.

Conclusion

Introduction

In our previous article, we learned how to factor the cubic expression $x^3 - 7x^2 - 5x + 35$ by grouping. We divided the terms into two groups, factored out common factors from each group, and then combined the factored groups. Finally, we factored out the common binomial factor to get the resulting expression.

Q&A

Q: What is the method of grouping in factorizing a cubic expression? A: The method of grouping involves dividing the terms of the expression into two groups, such that each group has two terms. We then factor out common factors from each group.

Q: How do we divide the terms into two groups? A: To divide the terms into two groups, we look for a way to split the expression into two parts, such that each part has two terms. In the case of the expression $x^3 - 7x^2 - 5x + 35$, we can divide the terms into two groups as follows:

(x3βˆ’7x2)βˆ’(5xβˆ’35)\left(x^3 - 7x^2\right) - \left(5x - 35\right)

Q: How do we factor out common factors from each group? A: To factor out common factors from each group, we look for a common factor that can be factored out from each term in the group. In the case of the first group, we can factor out $x^2$:

x2(xβˆ’7)x^2\left(x - 7\right)

From the second group, we can factor out $-5$:

βˆ’5(xβˆ’7)-5\left(x - 7\right)

Q: How do we combine the factored groups? A: To combine the factored groups, we multiply the factored groups together:

(x2βˆ’5)(xβˆ’7)(x^2 - 5)\left(x - 7\right)

Q: What is the resulting expression? A: The resulting expression is:

(x2βˆ’5)(xβˆ’7)(x^2 - 5)\left(x - 7\right)

Q: How does this compare with the other options? A: Let's compare our resulting expression with the other options:

  • Option A: $\left(x^2 - 7\right)(x - 5)$
  • Option B: $\left(x^2 - 7\right)(x + 5)$
  • Option C: $\left(x^2 - 5\right)(x - 7)$
  • Option D: $\left(x^2 + 5\right)(x - 7)$

Our resulting expression matches with Option C: $\left(x^2 - 5\right)(x - 7)$.

Common Mistakes

  • Not dividing the terms into two groups correctly: Make sure to divide the terms into two groups in a way that each group has two terms.
  • Not factoring out common factors correctly: Make sure to factor out common factors from each group correctly.
  • Not combining the factored groups correctly: Make sure to multiply the factored groups together correctly.

Conclusion

In this article, we answered some common questions about factorizing a cubic expression by grouping. We learned how to divide the terms into two groups, factor out common factors from each group, and then combine the factored groups. Finally, we factored out the common binomial factor to get the resulting expression.