Question

Use the grouping method to factor the polynomial below completely.
x3-5x2+3x-15
0(47+314-51
0 (x2-3)(x+5)
0(82+5)(4-3)
0 (x2-5)(x+3)

Answer 1

To factor the polynomial \( x^3 - 5x^2 + 3x - 15 \) using the grouping method, we will group the terms to factor by parts.

1. **Group the terms**:
\[
(x^3 - 5x^2) + (3x - 15)
\]

2. **Factor out the common factors in each group**:
- From the first group \( x^3 - 5x^2 \), we can factor out \( x^2 \):
\[
x^2(x - 5)
\]
- From the second group \( 3x - 15 \), we can factor out \( 3 \):
\[
3(x - 5)
\]

3. **Rewrite the polynomial**:
Combining these, we can write:
\[
x^2(x - 5) + 3(x - 5)
\]

4. **Factor out the common binomial factor** \( (x - 5) \):
\[
(x - 5)(x^2 + 3)
\]

5. **Final factored form**:
The polynomial \( x^3 - 5x^2 + 3x - 15 \) factors to:
\[
(x - 5)(x^2 + 3)
\]

Thus, the complete factorization using the grouping method is:
\[
\boxed{(x - 5)(x^2 + 3)}
\]

Note that \( x^2 + 3 \) cannot be factored further over the real numbers since it has no real roots.