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 start by grouping the terms. Here's the process step-by-step:

1. Group the terms:
   \( (2x^3 + x^2) + (-8x - 4) \)

2. Factor out the common terms in each group:
   From the first group \( 2x^3 + x^2 \), we can factor out \( x^2 \):
   \( x^2(2x + 1) \)
   From the second group \(-8x - 4\), we can factor out \(-4\):
   \( -4(2x + 1) \)

3. Rewrite the expression with the factors:
   Now we can write the polynomial as: \[ x^2(2x + 1) - 4(2x + 1) \]

4. Factor out the common factor \((2x + 1)\):
   We can factor out \((2x + 1)\) from the expression: \[ (2x + 1)(x^2 - 4) \]

5. Factor the remaining quadratic:
   The term \(x^2 - 4\) is a difference of squares, which can be factored as: \[ x^2 - 4 = (x - 2)(x + 2) \]

6. Combine everything:
   Putting it all together, we have: \[ (2x + 1)(x - 2)(x + 2) \]

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