Factors of the Polynomial x³ + 5x² + 2x – 8
To find the factors of the polynomial x³ + 5x² + 2x – 8, we can follow a systematic approach:
Step 1: Factorization Process
First, we’ll use the Rational Root Theorem to identify potential rational roots. The Rational Root Theorem suggests that any possible rational root, represented as p/q, is a factor of the constant term (-8) over a factor of the leading coefficient (1). Thus, the possible rational roots could be:
- ±1, ±2, ±4, ±8
Step 2: Testing Possible Roots
Now, we need to test these possible roots by substituting them into the polynomial:
- x = 1: 1³ + 5(1)² + 2(1) – 8 = 1 + 5 + 2 – 8 = 0
Therefore, x = 1 is a root. - x = -1: (-1)³ + 5(-1)² + 2(-1) – 8 = -1 + 5 – 2 – 8 = -6
Not a root. - x = 2: 2³ + 5(2)² + 2(2) – 8 = 8 + 20 + 4 – 8 = 24
Not a root. - x = -2: (-2)³ + 5(-2)² + 2(-2) – 8 = -8 + 20 – 4 – 8 = 0
Therefore, x = -2 is a root. - x = 4: 4³ + 5(4)² + 2(4) – 8 = 64 + 80 + 8 – 8 = 144
Not a root. - x = -4: (-4)³ + 5(-4)² + 2(-4) – 8 = -64 + 80 – 8 – 8 = 0
Therefore, x = -4 is a root. - x = 8: 8³ + 5(8)² + 2(8) – 8 = 512 + 320 + 16 – 8 = 840
Not a root. - x = -8: (-8)³ + 5(-8)² + 2(-8) – 8 = -512 + 320 – 16 – 8 = -216
Not a root.
Step 3: Polynomial Division
From our testing, we have identified potential roots at x = 1, x = -2, and x = -4. We will use synthetic division to factor the polynomial:
- Divide x³ + 5x² + 2x – 8 by (x – 1):
- The result gives us: x² + 6x + 8.
Step 4: Further Factorization
Now we will factor x² + 6x + 8:
- This can be factored as (x + 2)(x + 4).
Final Factorization
Putting it all together, we find:
- x³ + 5x² + 2x – 8 = (x – 1)(x + 2)(x + 4).
Conclusion
The factors of the polynomial x³ + 5x² + 2x – 8 are:
- (x – 1)
- (x + 2)
- (x + 4)
This process highlights the steps involved in determining the factors, incorporating both algebraic methods and polynomial root testing.