To find the factors of the polynomial x³ + 5x² + 2x + 8, we can start by looking for rational roots using the Rational Root Theorem. This theorem suggests that any possible rational root, p/q, would have p as a factor of the constant term (which is 8) and q as a factor of the leading coefficient (which is 1 here).
The factors of 8 are ±1, ±2, ±4, ±8. Since the leading coefficient is 1, we only consider these potential roots.
Let’s test these values by substituting them into the polynomial:
- For x = -1:
(-1)³ + 5(-1)² + 2(-1) + 8 = -1 + 5 – 2 + 8 = 10 (not a root) - For x = 1:
(1)³ + 5(1)² + 2(1) + 8 = 1 + 5 + 2 + 8 = 16 (not a root) - For x = -2:
(-2)³ + 5(-2)² + 2(-2) + 8 = -8 + 20 – 4 + 8 = 16 (not a root) - For x = 2:
(2)³ + 5(2)² + 2(2) + 8 = 8 + 20 + 4 + 8 = 40 (not a root) - For x = -4:
(-4)³ + 5(-4)² + 2(-4) + 8 = -64 + 80 – 8 + 8 = 16 (not a root) - For x = 4:
(4)³ + 5(4)² + 2(4) + 8 = 64 + 80 + 8 + 8 = 160 (not a root) - For x = -8:
(-8)³ + 5(-8)² + 2(-8) + 8 = -512 + 320 – 16 + 8 = -200 (not a root) - For x = 8:
(8)³ + 5(8)² + 2(8) + 8 = 512 + 320 + 16 + 8 = 856 (not a root)
As none of these test values yield a root, we can conclude that x³ + 5x² + 2x + 8 is irreducible over the rational numbers. Therefore, the polynomial doesn’t factor neatly into linear factors with integer coefficients.
However, it can still be analyzed as a whole. This polynomial can be factored into its irreducible form, or you could also solve it using numerical methods or graphing to find the approximate roots.
In summary, the polynomial x³ + 5x² + 2x + 8 does not have rational roots and thus cannot be factored further over the rationals.