Factor by grouping x3−4x2−4x+16 (1 point) Responses (x+4)(x+2) open paren x plus 4 close paren times open paren x plus 2 close paren (x+4)(x2−4) open paren x plus 4 close paren times open paren x squared minus 4 close paren (x−4)(x2−4) 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, we will group the terms and factor out common factors.

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

2. Factor out the common factors in each group:

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

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

4. Factor out the common binomial factor \( (x - 4) \): \[ (x - 4)(x^2 - 4) \]

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 to get the fully factored form: \[ (x - 4)(x - 2)(x + 2) \]

Given the options you provided, the expression factored by grouping is:

• \( (x - 4)(x - 2)(x + 2) \)

This answer is not one of the choices listed, but it is the correct completely factored form. Among the provided options, if we have to choose the most suitable, it would likely be:

• \( (x+4)(x+2) \) or \( (x−4)(x^2−4) \) — note that \( (x^2 - 4) \) can be used in more combinations.

Since \( (x - 4)(x^2 - 4) \) identifies part of the factorization, it is also correct for grouping.