Use factoring by grouping to solve the equation.

Question

n cubed minus 4 n squared minus n plus 4 equals 0
Question content area bottom
Part 1
nequals
  
enter your response here
​(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: \[ \boxed{4, 1, -1} \]