If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), find the remaining factors.

If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), find the remaining factors.

Answer Details

If \((x - 3)\) is a factor of \(2x^{3} + 3x^{2} - 17x - 30\), then we can use long division or synthetic division to divide the polynomial by \((x - 3)\) and find the remaining factors.

Using long division, we have:

          2x^2 + 9x + 10
    ------------------------
x - 3 | 2x^3 + 3x^2 - 17x - 30
         - (2x^3 - 6x^2)
         --------------
                  9x^2 - 17x
                  - (9x^2 - 27x)
                  -----------
                           10x - 30
                           - (10x - 30)
                           ---------
                                  0

The result of the division is \(2x^{2} + 9x + 10\), which is a quadratic polynomial. Therefore, the remaining factors are given by factoring this quadratic polynomial. We can factor it as follows:

\[2x^{2} + 9x + 10 = (2x + 5)(x + 2)\]

Therefore, the complete factorization of \(2x^{3} + 3x^{2} - 17x - 30\) is:

\[2x^{3} + 3x^{2} - 17x - 30 = (x - 3)(2x + 5)(x + 2)\]

So, the correct option is (2x + 5)(x + 2).