Factor by grouping x3−4x2−4x+16

Question

𝑥
3
−
4
𝑥
2
−
4
𝑥
+
16
(1 point)
Responses

(x+4)(x+2)
(
𝑥
+
4
)
(
𝑥
+
2
)
open paren x plus 4 close paren times open paren x plus 2 close paren

(x+4)(x2−4)
(
𝑥
+
4
)
(
𝑥
2
−
4
)
open paren x plus 4 close paren times open paren x squared minus 4 close paren

(x−4)(x2−4)
(
𝑥
−
4
)
(
𝑥
2
−
4
)
open paren x minus 4 close paren times open paren x squared minus 4 close paren

(x−4)(x−2)

Answer 1

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

1. Group the terms: \[ (x^3 - 4x^2) + (-4x + 16) \]

2. Factor out the common factors in each group:

   • From the first group \( x^3 - 4x^2 \), factor out \( x^2 \): \[ x^2(x - 4) \]
   • From the second group \( -4x + 16 \), factor out \(-4\): \[ -4(x - 4) \]

3. Combine the groups: This gives us: \[ x^2(x - 4) - 4(x - 4) \] Now, we can factor out the common term \( (x - 4) \): \[ (x - 4)(x^2 - 4) \]

4. Recognize and factor the difference of squares: Since \( x^2 - 4 \) can be factored further as \( (x + 2)(x - 2) \) (since \( x^2 - 4 = (x - 2)(x + 2) \)), we have: \[ (x - 4)(x + 2)(x - 2) \]

So, the factorization of \( x^3 - 4x^2 - 4x + 16 \) is: \[ (x - 4)(x + 2)(x - 2) \]

Given the response options, we can see none match exactly. However, the factorization provided above is correct, and it represents the polynomial as completely factored. If the choices were slightly altered to include this expanded version, it would have been more aligned.