To factor the polynomial x³ + 5x² + 6x – 30 by grouping, we can start by rearranging the terms to make grouping easier. First, we can group the first two terms and the last two terms:
(x³ + 5x²) + (6x – 30)
Next, we factor out the common factor from each group:
x²(x + 5) + 6(x – 5)
However, notice we can’t factor out the last part quite matches since we need the same factor. Recall we need to use synthetic or polynomial division to find possible rational roots. By trying out possible roots (like ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30), we find that x = 2 is a root.
Using synthetic division with x-2, we can divide the cubic polynomial:
- Using long division or synthetic division, we’ll express:
- x³ + 5x² + 6x – 30 = (x – 2)(x² + 7x + 15)
Finally, we check the quadratic factor x² + 7x + 15. This quadratic doesn’t factor nicely with integers since its discriminant (b² – 4ac) results in a negative number. Therefore, we conclude that the final factored form of the polynomial x³ + 5x² + 6x – 30 is:
(x – 2)(x² + 7x + 15)