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 x=2

I have no idea how to start this problem!!

You can check to see that x=-2 checks.

-8+56-48 does indeed equal zero, so x+2 is a factor.

Now divide x+2 into the polynomial (you are asked to use synthetic division) to get the remaining quadratic. Once you have that, you can factor to get the other two roots.

To use synthetic division to show that x=2 is a solution of the polynomial equation x^3 - 28x - 48 = 0, we can follow these steps:

Step 1: Set up the synthetic division table by writing down the coefficients of the polynomial equation in descending order of powers of x.

1 0 -28 -48

Step 2: Since we are testing if x=2 is a solution, write 2 on the left-hand side of the table, outside the table's enclosure.

Step 3: Draw a vertical line from the 2 in the left-hand side column down to the bottom of the table.

Step 4: Now, perform the synthetic division by distributing the 2 to each of the coefficients and adding column-wise.

1 0 -28 -48
_____________________
2 | 1 0 -28 -48
| 2 4 -48
_____________________
1 2 -24 -48

The result of the synthetic division is 1 2 -24 -48. The last entry in the bottom row (-48) represents the remainder of the division. If the remainder is 0, it means that x=2 is a solution of the polynomial equation, and the polynomial can be factored completely.

Since the remainder is not zero (-48), x=2 is not a solution, and we need to find other solutions or factor the polynomial differently.

To find the remaining solutions or factor the polynomial completely, we can use the Rational Root Theorem or factor theorem. However, in this case, the polynomial can be factored using other methods such as grouping or factoring by guess and check.

The factored form of the given polynomial x^3 - 28x - 48 = 0 is as follows:

(x - 4)(x + 4)(x + 6) = 0

So, the complete factorization of the polynomial is (x - 4)(x + 4)(x + 6) = 0.

The real zeros of the function are the values that make the polynomial equation equal to zero. From the factorization, we find three real zeros:

x = 4, -4, -6.

To use synthetic division, we need to rewrite the polynomial equation in the form (x - c)(ax^2 + bx + d) = 0, where c is the given value of x. In this case, we have x = 2.

Step 1: Set up the synthetic division. Write the coefficients of the polynomial equation in descending order of powers of x. The coefficients of x^3, x^2, x, and the constant term will form the rows.

| 1 0 -28 -48
|_________________
2 |

Step 2: Draw a horizontal line beneath the coefficients. Bring down the first coefficient (1) directly below the line.

| 1 0 -28 -48
|_________________
2 | 1

Step 3: Multiply the value at the top of the column by the divisor (2) and write the result beneath the next coefficient (0).

| 1 0 -28 -48
|_________________
2 | 1
2

Step 4: Add the values in the current column. Write the sum beneath the next coefficient (-28).

| 1 0 -28 -48
|_________________
2 | 1
2
-28

Step 5: Repeat steps 3 and 4 until all the coefficients have been processed.

| 1 0 -28 -48
|_________________
2 | 1
2
-28
4
-48

Step 6: The number at the bottom is the remainder. If the remainder is equal to zero, then x = 2 is indeed a solution.

In this case, the remainder is -48, which is not equal to zero. Therefore, x = 2 is not a solution to the third-degree polynomial equation. We can conclude that the polynomial is not evenly divisible by (x - 2).

Since x = 2 is not a zero, we need to find the remaining zeros of the function.

To do this, we can find the zeros using the Rational Root Theorem or by using a graphing calculator. Let's use the Rational Root Theorem.

The Rational Root Theorem states that if a polynomial equation has a rational root p/q, where p is a factor of the constant term (-48) and q is a factor of the leading coefficient (1), then it will satisfy the equation.

The factors of the constant term (-48) are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, and ±48.

The factors of the leading coefficient (1) are ±1.

Now, test each possible root using synthetic division until you find a zero.

Let's test:
- p = 1, q = 1: Trying 1/1 = 1.
Using synthetic division, we get a remainder of -24, which is not zero.

- p = 1, q = -1: Trying 1/-1 = -1.
Using synthetic division, we get a remainder of 52, which is not zero.

- p = 2, q = 1: Trying 2/1 = 2.
Using synthetic division, we get a remainder of 0, which means x = 2 is a zero.

Now, to factor the polynomial completely, we know that (x - 2) is a factor since x = 2 is a zero. We can use synthetic division again to divide the polynomial by (x - 2) to find the remaining factor.

| 1 0 -28
|_________________
2 | 1
2
-28
4

After the division, we obtain the following:

x^2 + 2x - 24 = 0

Now, we can factor the remaining quadratic equation further.

(x^2 + 2x - 24) = (x - 4)(x + 6)

Therefore, the factored form of the polynomial is:

(x - 2)(x - 4)(x + 6) = 0

The real zeros of the function are x = 2, x = 4, and x = -6.