Factoring the following polynomial expression completely:

Question

2
x
3
+
x
2
−
8
x
−
4
2x
3
+x
2
−8x−4

Answer 1

To factor the polynomial \( 2x^3 + x^2 - 8x - 4 \) completely, we can use grouping.

1. First, we group the terms: \[ (2x^3 + x^2) + (-8x - 4) \]

2. Now factor out the common factors in each group: \[ x^2(2x + 1) - 4(2x + 1) \]

3. We see that \( (2x + 1) \) is a common factor: \[ (2x + 1)(x^2 - 4) \]

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

5. Combining all parts, the complete factorization is: \[ (2x + 1)(x - 2)(x + 2) \]

Thus, the complete factorization of the polynomial \( 2x^3 + x^2 - 8x - 4 \) is: \[ \boxed{(2x + 1)(x - 2)(x + 2)} \]