Question

Factor by grouping x3−4x2−4x+16

Answer 1

To factor the polynomial \( x^3 - 4x^2 - 4x + 16 \) by grouping, we can follow these steps:

1. **Group the terms**: We can split the polynomial into two groups:
\[
(x^3 - 4x^2) + (-4x + 16)
\]

2. **Factor out the common factor from each group**:
- In the first group \( x^3 - 4x^2 \), the common factor is \( x^2 \):
\[
x^2(x - 4)
\]
- In the second group \( -4x + 16 \), the common factor is \(-4\):
\[
-4(x - 4)
\]

3. **Combine the factored groups**:
Now we can rewrite the expression using the factored form:
\[
x^2(x - 4) - 4(x - 4)
\]

4. **Factor out the common binomial factor**:
Since both terms contain the factor \((x - 4)\), we can factor that out:
\[
(x - 4)(x^2 - 4)
\]

5. **Notice that \(x^2 - 4\) is a difference of squares, which can be factored further**:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

6. **Combine everything**:
Putting it all together, we have:
\[
(x - 4)(x - 2)(x + 2)
\]

Thus, the factored form of the polynomial \( x^3 - 4x^2 - 4x + 16 \) is:
\[
\boxed{(x - 4)(x - 2)(x + 2)}
\]