Question

Factoring the following polynomial expression completely:
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:
\[
\boxed{(2x + 1)(x - 2)(x + 2)}
\]