Question

Factor completely.
2, x, cubed, plus, 5, x, squared, minus, 28, x, minus, 70
2x^3+5x^2−28x−70

Answer 1

To factor the expression \(2x^3 + 5x^2 - 28x - 70\) completely, we can follow these steps.

1. **Group the terms**: We can group the terms into two pairs:
\[
(2x^3 + 5x^2) + (-28x - 70)
\]

2. **Factor out common factors from each pair**:
In the first group \(2x^3 + 5x^2\), the common factor is \(x^2\):
\[
x^2(2x + 5)
\]

In the second group \(-28x - 70\), we can factor out \(-14\):
\[
-14(2x + 5)
\]

3. **Combine the factored terms**:
Now we can rewrite the entire expression:
\[
x^2(2x + 5) - 14(2x + 5)
\]

Notice that \((2x + 5)\) is a common factor:
\[
(2x + 5)(x^2 - 14)
\]

4. **Factor \(x^2 - 14\)**: The expression \(x^2 - 14\) can be factored further as a difference of squares:
\[
x^2 - 14 = x^2 - \sqrt{14}^2 = (x - \sqrt{14})(x + \sqrt{14})
\]

5. **Write the complete factored form**:
Putting it all together, the complete factorization of the original expression \(2x^3 + 5x^2 - 28x - 70\) is:
\[
(2x + 5)(x - \sqrt{14})(x + \sqrt{14})
\]

Hence, the complete factorization is:
\[
(2x + 5)(x - \sqrt{14})(x + \sqrt{14})
\]