Question 1209715
Here's how to find the third root and the values of *a* and *c*:

1. **Use the factor theorem:** Since 1/2 and 7 are roots of P(x), then (x - 1/2) and (x - 7) are factors of P(x).  Therefore, we can write P(x) as:

   P(x) = 2(x - 1/2)(x - 7)(x - r)

   where 'r' is the third root. The factor of 2 is included so that the coefficient of the x³ term matches the given polynomial.

2. **Expand the factored form:**
   P(x) = 2(x - 1/2)(x - 7)(x - r)
   P(x) = (2x - 1)(x - 7)(x - r)
   P(x) = (2x² - 15x + 7)(x - r)
   P(x) = 2x³ - 15x² + 7x - 2rx² + 15rx - 7r
   P(x) = 2x³ + (-15 - 2r)x² + (7 + 15r)x - 7r

3. **Compare coefficients:** Now, compare the coefficients of the expanded form with the given form P(x) = 2x³ + ax² - 23x + c:

   * Coefficient of x²:  a = -15 - 2r
   * Coefficient of x: -23 = 7 + 15r
   * Constant term: c = -7r

4. **Solve for r:** From the coefficient of x, we can solve for r:

   -23 = 7 + 15r
   -30 = 15r
   r = -2

5. **Find a and c:** Now that we know r = -2, we can find *a* and *c*:

   a = -15 - 2(-2) = -15 + 4 = -11
   c = -7(-2) = 14

Therefore, the third root is -2, and the ordered pair (a, c) is (-11, 14).