Question

Let R,S,T, and V be the roots of

f(x)=2x⁴ - 3x³ -24x² + 13x + 12.
If f(-1/2) = 0 and if (x-1) is a factor of f(x), find the product RSTV.

Answer 1

To find the product RSTV, divide the given polynomial by (x-1) using synthetic division. The remaining roots can be found by solving the quadratic equation. The product RSTV is 6.

Step-by-step explanation:

The given polynomial is f(x) = 2x⁴ - 3x³ - 24x² + 13x + 12. Given that f(-1/2) = 0 and (x-1) is a factor of f(x), we can find the remaining roots by dividing f(x) by (x-1) using synthetic division. The quotient will be a quadratic equation, and we can solve it to find the remaining roots.

Using synthetic division, we find that the quadratic equation is 2x² - 5x - 12 = 0.

Factoring this quadratic equation, we get (2x + 3)(x - 4) = 0. So, the remaining roots are x = -3/2 and x = 4. Therefore, the product RSTV is equal to -1/2 * -3/2 * 1 * 4 = 6.