X4-4x3-2x2+12x+9
To work with the polynomial \( x^4 - 4x^3 - 2x^2 + 12x + 9 \), we might want to perform several operations like factoring, finding roots, or evaluating the polynomial for particular values of \( x \).
1. Factoring
To factor the polynomial, one way is to look for rational roots using the Rational Root Theorem. The theorem suggests that any rational root, in the form of \( \frac{p}{q} \), involves \( p \) (factors of the constant term, 9) and \( q \) (factors of the leading coefficient, 1).
Possible rational roots to test are \( \pm 1, \pm 3, \pm 9 \).
Testing potential roots:
For \( x = 1 \): \[ 1^4 - 4(1^3) - 2(1^2) + 12(1) + 9 = 1 - 4 - 2 + 12 + 9 = 16 \quad (\text{not a root}) \]
For \( x = -1 \): \[ (-1)^4 - 4(-1)^3 - 2(-1)^2 + 12(-1) + 9 = 1 + 4 - 2 - 12 + 9 = 0 \quad (\text{is a root}) \]
Now that we have found a root \( x = -1 \), we can factor the polynomial by dividing.
2. Synthetic Division
To factor by synthetic division:
-1 | 1 -4 -2 12 9
| -1 5 -3 -9
---------------------------
1 -5 3 9 0
The quotient is \( x^3 - 5x^2 + 3x + 9 \).
Now we can rewrite the polynomial: \[ x^4 - 4x^3 - 2x^2 + 12x + 9 = (x + 1)(x^3 - 5x^2 + 3x + 9) \]
3. Finding further roots
Next, we can attempt to factor \( x^3 - 5x^2 + 3x + 9 \). We can again use the Rational Root Theorem on \( \pm 1, \pm 3, \pm 9 \).
For \( x = -3 \): \[ (-3)^3 - 5(-3)^2 + 3(-3) + 9 = -27 - 45 - 9 + 9 = -72 \quad (\text{not a root}) \]
For \( x = 3 \): \[ 3^3 - 5(3^2) + 3(3) + 9 = 27 - 45 + 9 + 9 = 0 \quad (\text{is a root}) \]
Now, do synthetic division again with \( x = 3 \):
3 | 1 -5 3 9
| 3 -6 -9
---------------------
1 -2 -3 0
The factorization gives: \[ x^3 - 5x^2 + 3x + 9 = (x - 3)(x^2 - 2x - 3) \]
4. Factoring \( x^2 - 2x - 3 \):
Now, we can factor \( x^2 - 2x - 3 \): \[ x^2 - 2x - 3 = (x - 3)(x + 1) \]
Final Factored Form
Combining all the factors: \[ x^4 - 4x^3 - 2x^2 + 12x + 9 = (x + 1)^2 (x - 3)^2 \]
Summary
The original polynomial \( x^4 - 4x^3 - 2x^2 + 12x + 9 \) factors to: \[ (x + 1)^2 (x - 3)^2 \]
