SOLUTION: Kuta Software Infinite Algebra 2: Factoring by Grouping ... - Free Printable

The image shows a worksheet titled "Factoring By Grouping" from *Kuta Software - Infinite Algebra 2*. The task is to factor each polynomial completely using the grouping method. There are six problems, and all of them have been partially worked out with correct answers boxed.

Let’s go through each problem step-by-step to verify the solutions and explain the reasoning behind factoring by grouping.

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🔷 Factoring by Grouping: General Strategy

This method is used for polynomials with four or more terms. The steps are:

1. Group the terms into two pairs.
2. Factor out the GCF (greatest common factor) from each pair.
3. If the resulting binomials are the same, factor out the common binomial.
4. Write the final factored form.

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## ✔ Problem 1:
$ 12a^3 - 9a^2 + 4a - 3 $

Step 1: Group

$$
(12a^3 - 9a^2) + (4a - 3)
$$

Step 2: Factor GCF from each group

- $ 3a^2(4a - 3) + 1(4a - 3) $

Step 3: Factor out common binomial

$$
(4a - 3)(3a^2 + 1)
$$

✔ Correct!

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## ✔ Problem 2:
$ 2p^3 + 5p^2 + 6p + 15 $

Step 1: Group

$$
(2p^3 + 5p^2) + (6p + 15)
$$

Step 2: Factor GCF

- $ p^2(2p + 5) + 3(2p + 5) $

Step 3: Common binomial

$$
(2p + 5)(p^2 + 3)
$$

✔ Correct!

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## ✔ Problem 3:
$ 3n^3 - 4n^2 + 9n - 12 $

Step 1: Group

$$
(3n^3 - 4n^2) + (9n - 12)
$$

Step 2: Factor GCF

- $ n^2(3n - 4) + 3(3n - 4) $

Step 3: Common binomial

$$
(3n - 4)(n^2 + 3)
$$

Note: The answer written is $(n^2 + 3)(3n - 4)$ — same thing, just order swapped.

✔ Correct!

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## ✔ Problem 4:
$ 12n^3 + 4n^2 + 3n + 1 $

Step 1: Group

$$
(12n^3 + 4n^2) + (3n + 1)
$$

Step 2: Factor GCF

- $ 4n^2(3n + 1) + 1(3n + 1) $

Step 3: Common binomial

$$
(3n + 1)(4n^2 + 1)
$$

✔ Correct!

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## ✔ Problem 5:
$ m^3 - m^2 + 2m - 2 $

Step 1: Group

$$
(m^3 - m^2) + (2m - 2)
$$

Step 2: Factor GCF

- $ m^2(m - 1) + 2(m - 1) $

Step 3: Common binomial

$$
(m - 1)(m^2 + 2)
$$

✔ Correct!

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## ✔ Problem 6:
$ 5n^3 - 10n^2 + 3n - 6 $

Step 1: Group

$$
(5n^3 - 10n^2) + (3n - 6)
$$

Step 2: Factor GCF

- $ 5n^2(n - 2) + 3(n - 2) $

Step 3: Common binomial

$$
(n - 2)(5n^2 + 3)
$$

✔ Correct!

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## 📌 Summary of All Answers:

| Problem | Factored Form |
|--------|----------------|
| 1 | $ (4a - 3)(3a^2 + 1) $ |
| 2 | $ (2p + 5)(p^2 + 3) $ |
| 3 | $ (3n - 4)(n^2 + 3) $ |
| 4 | $ (3n + 1)(4n^2 + 1) $ |
| 5 | $ (m - 1)(m^2 + 2) $ |
| 6 | $ (n - 2)(5n^2 + 3) $ |

All the answers shown in the image are correct and properly factored by grouping.

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## ✔ Final Notes:
- This method works best when you can split the polynomial into two groups that share a common binomial factor.
- Always check your work by expanding the factored form to see if you get back the original expression.

Let me know if you'd like a visual explanation or help with a different type of factoring!