Which Shows One Way To Determine The Factors Of $x^3-9x^2+5x-45$ By Grouping?A. $x^2(x-9)-5(x-9)$B. \$x^2(x+9)-5(x+9)$[/tex\]C. $x(x^2+5)-9(x^2+5)$D. $x(x^2-5)-9(x^2-5)$

by ADMIN 179 views

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 the polynomial to facilitate factoring. In this article, we will explore how to determine the factors of a given polynomial by grouping.

What is Grouping in Factoring?

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 with common factors. By grouping, we can simplify the polynomial and make it easier to factor.

Example: Factoring by Grouping

Let's consider the polynomial $x3-9x2+5x-45$. Our goal is to determine the factors of this polynomial by grouping. To do this, we need to rearrange the terms of the polynomial to create two or more groups of terms that can be factored separately.

Step 1: Rearrange the Terms

The first step in factoring by grouping is to rearrange the terms of the polynomial. We can do this by grouping the terms that have common factors. In this case, we can group the first two terms and the last two terms:

x3−9x2+5x−45=(x3−9x2)+(5x−45)x^3-9x^2+5x-45 = (x^3-9x^2)+(5x-45)

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 x2x^2:

(x3−9x2)=x2(x−9)(x^3-9x^2) = x^2(x-9)

In the second group, we can factor out 5:

(5x−45)=5(x−9)(5x-45) = 5(x-9)

Step 3: Combine the Groups

Now that we have factored out common factors from each group, we can combine the groups to get the final factored form of the polynomial:

x2(x−9)−5(x−9)x^2(x-9)-5(x-9)

Conclusion

In this article, we have seen how to determine the factors of a polynomial by grouping. By rearranging the terms of the polynomial and factoring out common factors, we can simplify the polynomial and make it easier to factor. The example we used in this article shows how to factor the polynomial $x3-9x2+5x-45$ by grouping.

Which Option is Correct?

Now that we have factored the polynomial by grouping, we can compare our result with the options given in the problem. The correct option is:

A. $x^2(x-9)-5(x-9)$

This option matches the factored form of the polynomial that we obtained by grouping.

Why is Option A Correct?

Option A is correct because it matches the factored form of the polynomial that we obtained by grouping. The polynomial $x^2(x-9)-5(x-9)$ is equivalent to the original polynomial $x3-9x2+5x-45$, and it can be factored further to obtain the complete factorization of the polynomial.

What are the Other Options?

The other options given in the problem are:

B. $x^2(x+9)-5(x+9)$

C. $x(x2+5)-9(x2+5)$

D. $x(x2-5)-9(x2-5)$

These options do not match the factored form of the polynomial that we obtained by grouping, and they are therefore incorrect.

Why are Options B, C, and D Incorrect?

Options B, C, and D are incorrect because they do not match the factored form of the polynomial that we obtained by grouping. The polynomial $x^2(x+9)-5(x+9)$, $x(x2+5)-9(x2+5)$, and $x(x2-5)-9(x2-5)$ are not equivalent to the original polynomial $x3-9x2+5x-45$, and they cannot be factored further to obtain the complete factorization of the polynomial.

Conclusion

Introduction

In our previous article, we explored how to determine the factors of a polynomial by grouping. We saw how to rearrange the terms of the polynomial to create two or more groups of terms that can be factored separately. In this article, we will answer some frequently asked questions about determining factors of a polynomial by grouping.

Q: What is the first step in factoring a polynomial by grouping?

A: The first step in factoring a polynomial by grouping is to rearrange the terms of the polynomial to create two or more groups of terms that can be factored separately.

Q: How do I know which terms to group together?

A: To determine which terms to group together, look for common factors among the terms. Group the terms that have common factors together.

Q: What if I have multiple groups of terms? How do I factor them?

A: If you have multiple groups of terms, factor out common factors from each group separately. Then, combine the groups to get the final factored form of the polynomial.

Q: Can I always factor a polynomial by grouping?

A: No, not all polynomials can be factored by grouping. However, this method is particularly useful when the polynomial has multiple terms with common factors.

Q: What if I get stuck while factoring a polynomial by grouping?

A: If you get stuck while factoring a polynomial by grouping, try rearranging the terms of the polynomial again. Look for different ways to group the terms together.

Q: Can I use other factoring methods in addition to grouping?

A: Yes, you can use other factoring methods in addition to grouping. Some common factoring methods include factoring out the greatest common factor (GCF), factoring by difference of squares, and factoring by sum and difference.

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 to get the original polynomial. If the product is equal to the original polynomial, then you have factored it correctly.

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

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

  • Not rearranging the terms of the polynomial correctly
  • Not factoring out common factors from each group
  • Not combining the groups correctly
  • Not checking the product of the factors to ensure it is equal to the original polynomial

Conclusion

In conclusion, determining factors of a polynomial by grouping is a useful technique that can help simplify complex expressions and solve equations. By following the steps outlined in this article, you can master this technique and become proficient in factoring polynomials by grouping.

Practice Problems

To practice factoring polynomials by grouping, try the following problems:

  1. Factor the polynomial $x3+8x2+15x+6$ by grouping.
  2. Factor the polynomial $x^2-4x+3$ by grouping.
  3. Factor the polynomial $x3-2x2-5x+10$ by grouping.

Answer Key

  1. (x3+8x2)+(15x+6)=x2(x+8)+3(x+8)=(x2+3)(x+8)(x^3+8x^2)+(15x+6) = x^2(x+8)+3(x+8) = (x^2+3)(x+8)

  2. (x2−4x)+(3)=x(x−4)+3=(x−3)(x−1)(x^2-4x)+(3) = x(x-4)+3 = (x-3)(x-1)

  3. (x3−2x2)−5x+10=x2(x−2)−5(x−2)=(x2−5)(x−2)(x^3-2x^2)-5x+10 = x^2(x-2)-5(x-2) = (x^2-5)(x-2)