Final answer:
The missing values in the factored polynomial expression are found by matching coefficients and solving a system of equations. The first step reveals that a is 2, and then by comparing coefficients of x^2 and x, a system of equations is established to find b and c.
Step-by-step explanation:
To find the missing values of a, b, and c in the factored polynomial 2x^3-2x^2-24x=ax(x+b)(x+c), we can compare the given polynomial to the factored form and match coefficients.
First, notice that the polynomial is already factored on the right-hand side of the equation. Since the leading term on the left side is 2x^3, the value of a must be 2 to match the leading coefficient. Next, we can expand the right side to get 2x^3 + (2b + 2c)x^2 + 2bcx. Comparing this with the original polynomial, 2x^3 - 2x^2 - 24x, we can set up equations based on the coefficients of x^2 and x:
- For x^2: 2b + 2c = -2
- For x: 2bc = -24
To find b and c, we solve this system of equations. Dividing the first equation by 2 gives us b + c = -1. Now, you can plug this into the second equation, assuming b multiplied by c equals -12 (after dividing by 2). By solving these equations, you will find the values of b and c.