This is the question: "Find the values of b and c for which x^2+bx+c is a factor of 2x^3-3x^2-3x-5." I was suggested to use long division, but how do i do long division when i don't know what b and c are??

To find the values of b and c for which x^2+bx+c is a factor of 2x^3-3x^2-3x-5, you can use synthetic division instead of long division. Synthetic division allows you to divide polynomials without explicitly knowing the values of b and c.

Here's how you can use synthetic division to find the desired values:

Step 1: Set up the synthetic division. Write the coefficients of the polynomial you want to divide by (x^2+bx+c) and the dividend (2x^3-3x^2-3x-5). Make sure the terms involving x are in decreasing order of degree. In this case, the divisor is x^2+bx+c and the dividend is 2x^3-3x^2-3x-5.

Step 2: The divisor is in the form of x^2+bx+c, which means it can be factored as (x-r)(x-s), where r and s are the roots of the quadratic equation. Recall that the roots of the quadratic equation can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are the coefficients of the quadratic equation.

Step 3: Set up two synthetic divisions, one for each factor. Use the values you found for r and s in Step 2, which will be the first numbers in each synthetic division. Perform the synthetic division by dividing the dividend by each factor (x-r) and (x-s).

Step 4: Analyze the results. If the divisor (x^2+bx+c) is a factor of the polynomial, the synthetic division will yield a remainder of zero in both cases. Therefore, you need to find the values of b and c that make both synthetic divisions have a remainder of zero.

Step 5: Solve for b and c. By examining the synthetic divisions, you can find the values of b and c that make the remainders zero. These values will give you the desired factors.

By following this process, you can find the values of b and c for which x^2+bx+c is a factor of 2x^3-3x^2-3x-5, without explicitly knowing the values of b and c beforehand.