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