Final answer:
To find all factors of the polynomial 2x³ + 3x² - 4x - 2, divide it by (x-1) to obtain the quotient 2x² + 5x + 2, which factors further into (2x + 1)(x + 2). Thus, the complete factorization of the polynomial is (x - 1)(2x + 1)(x + 2).
Step-by-step explanation:
To find all factors of the polynomial 2x³ +3x² - 4x - 2, given that (x-1) is a factor, we need to perform polynomial long division or use synthetic division to divide the polynomial by (x-1).
- Set up the division:
- Divide the first term of the polynomial by the first term of the divisor, and multiply the divisor by that quotient. Subtract the result from the polynomial.
- Continue the process until you reach a remainder of 0.
- This will give you a quotient of 2x² + 5x + 2.
Now, we look for factors of the quadratic equation 2x² + 5x + 2. This can be factored into (2x + 1)(x + 2).
The original polynomial can thus be expressed as the product of its factors: (x - 1)(2x + 1)(x + 2).