Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all the real zeros of the function.

x^3 - 28x - 48 = 0

Value of x = -4

Please help!!Thank you

Nevermind I think I figured it out.

To use synthetic division to show that x = -4 is a solution of the given polynomial equation, we need to perform the following steps:

Step 1: Set up the synthetic division table. Place the coefficients of the polynomial equation in descending order in the table:

-4 | 1 0 -28 -48

Step 2: Bring down the first coefficient (1) to the bottom row:

-4 | 1

Step 3: Multiply -4 by the number in the bottom row (1) and write the result under the next coefficient, which is 0:

-4 | 1
-4

Step 4: Add the two numbers in the bottom row (1 + (-4) = -3), and write the result under the next coefficient, which is -28:

-4 | 1 | -3
-4
_________
-31

Step 5: Repeat the previous steps until you reach the last coefficient, -48:

-4 | 1 | -3 | -31
-4 | 28
__________
-3 | -3

The remainder is -3.

If the remainder is zero, it means that x = -4 is a solution of the polynomial equation. Since the remainder is not zero, x = -4 is not a solution.

To factor the polynomial completely, we can use the results obtained from synthetic division and write the polynomial equation as:

(x^3 - 28x - 48) = (x + 4)(x^2 - 4x - 12)

The quadratic factor, x^2 - 4x - 12, can be factored further as:

(x + 4)(x - 6)(x + 2)

The real zeros of the function are the values of x that satisfy the equation (x + 4)(x - 6)(x + 2) = 0:

x = -4, x = 6, x = -2