Question

Use factoring by grouping to solve the equation.
n cubed minus 4 n squared minus n plus 4 equals 0
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Part 1
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​(Simplify your answer. Use a comma to separate answers as​ needed.)

Answer 1

To solve the equation \( n^3 - 4n^2 - n + 4 = 0 \) using factoring by grouping, we can follow these steps:

1. **Group the terms**:
\[
(n^3 - 4n^2) + (-n + 4)
\]

2. **Factor out the common factors in each group**:
From the first group \( n^3 - 4n^2 \), we can factor out \( n^2 \):
\[
n^2(n - 4)
\]
From the second group \(-n + 4\), we can factor out \(-1\):
\[
-1(n - 4)
\]

Now we can write:
\[
n^2(n - 4) - 1(n - 4)
\]

3. **Factor out the common binomial \((n - 4)\)**:
\[
(n - 4)(n^2 - 1)
\]

4. **Recognize that \( n^2 - 1 \) is a difference of squares and factor it**:
\[
n^2 - 1 = (n - 1)(n + 1)
\]

So the entire equation factored is:
\[
(n - 4)(n - 1)(n + 1) = 0
\]

5. **Set each factor equal to zero**:
\[
n - 4 = 0 \quad \Rightarrow \quad n = 4
\]
\[
n - 1 = 0 \quad \Rightarrow \quad n = 1
\]
\[
n + 1 = 0 \quad \Rightarrow \quad n = -1
\]

6. **Final answers**:
The solutions to the equation \( n^3 - 4n^2 - n + 4 = 0 \) are:
\[
n = 4, 1, -1
\]

Thus, the answer in simplified form is:
\[
\boxed{4, 1, -1}
\]