Final answer:
The remaining factors of the polynomial p(x) = 2x^3 + 13x^2 + 17x - 12, aside from (x+4), can be found by performing synthetic division using the root x = -4 and then factoring the resulting quadratic polynomial or using the quadratic formula if necessary.
Step-by-step explanation:
To find the remaining factors of the polynomial p(x) = 2x^3 + 13x^2 + 17x - 12, given that (x+4) is a factor, we must perform polynomial long division or synthetic division. We can use the root associated with the factor, which is x = -4, to apply synthetic division.
First, we write down the coefficients of p(x): 2, 13, 17, -12. Placing -4 to the left side, we perform the synthetic division process:
- Bring down the first coefficient, which is 2.
- Multiply -4 by 2 and write the result under 13.
- Add that result to 13, put the sum below, and multiply by -4 again, continuing this process until the end.
Upon completion, we will have the coefficients of the quotient polynomial. The result of this division will provide us with a second degree polynomial whose factors we can then determine by either factoring or using the quadratic formula, if necessary.
If we identify two more real roots, one can write these roots in the form of factors such as (x - root1)(x - root2), thus providing the remaining factors of the given polynomial.