Factor by grouping: $$ x^{3}-3 x^{2}-4 x+12 $$

Question

UpStudy AI · Accepted Answer

To factor the polynomial $$ x^{3} - 3x^{2} - 4x + 12 $$ by grouping, follow these steps:

1. **Group the terms:**
   $$
   (x^{3} - 3x^{2}) + (-4x + 12)
   $$

2. **Factor out the greatest common factor (GCF) from each group:**
   $$
   x^{2}(x - 3) - 4(x - 3)
   $$

3. **Factor out the common binomial factor $$(x - 3)$$:**
   $$
   (x - 3)(x^{2} - 4)
   $$

4. **Factor the difference of squares in the second factor:**
   $$
   x^{2} - 4 = (x - 2)(x + 2)
   $$

5. **Combine all factors:**
   $$
   (x - 3)(x - 2)(x + 2)
   $$

**Final Factored Form:**
$$
(x - 3)(x - 2)(x + 2)
$$

**Verification:**

To ensure the factorization is correct, you can expand the factors:

$$
(x - 3)(x - 2)(x + 2) = (x - 3)(x^2 - 4) = x^3 - 3x^2 - 4x + 12
$$

Which matches the original polynomial.

**Answer:** $$(x - 3)(x - 2)(x + 2)$$
$$(x - 3)(x - 2)(x + 2)$$

Factor by grouping: \( x^{3}-3 x^{2}-4 x+12 \)