One of the zeros of the function f(x)=x^3-28x-48 is x=-4. Factor the function. Show your work.

How do I solve this?? Please help??

If x=-4 is one of the solutions, then x+4 must be a factor of x^3 - 28x - 48

so do a long division. You must get a quadratic with no remainder.
solve that quadratic = 0 to get the other two zeros.

hint: the remaining quadratic factors to give 2 more integer solutions.

To factor the function f(x) = x^3 - 28x - 48, we will use the fact that x = -4 is a zero of the function.

Step 1: Use synthetic division to divide f(x) by (x + 4).

-4 | 1 0 -28 -48
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Step-by-Step Solution:
To divide f(x) by (x + 4), we use synthetic division as follows:

First, arrange the coefficients of the function in descending order:
1 0 -28 -48

Start the synthetic division by bringing down the first coefficient, which is 1, to the bottom row:
-4 | 1 0 -28 -48
---------------------
1

Multiply the number at the bottom row (1) by the divisor (-4), then write the result in the second column:
-4 | 1 0 -28 -48
------
1

Add the numbers in the second column:
-4 | 1 0 -28 -48
------
1

Bring down the next coefficient, which is 0, and add it to the number in the third column:
-4 | 1 0 -28 -48
------
1

Multiply the number at the bottom row (1) by the divisor (-4), then write the result in the third column:
-4 | 1 0 -28 -48
------
1 4

Add the numbers in the third column:
-4 | 1 0 -28 -48
------
1 4

Bring down the next coefficient, which is -28, and add it to the number in the fourth column:
-4 | 1 0 -28 -48
------
1 4

Multiply the number at the bottom row (1) by the divisor (-4), then write the result in the fourth column:
-4 | 1 0 -28 -48
------
1 4 112

Add the numbers in the fourth column:
-4 | 1 0 -28 -48
------
1 4 112

The numbers in the last row represent the coefficients of the quotient. In this case, the quotient is 1x^2 + 4x + 112.

Step 2: Write the factored form of the function.

From the synthetic division, we have:

f(x) = (x + 4)(x^2 + 4x + 112)

The factored form of the function f(x) = x^3 - 28x - 48 is (x + 4)(x^2 + 4x + 112).

To factor the given function f(x) = x^3 - 28x - 48, you can use synthetic division to divide the polynomial by the given zero x = -4. Here's how you can do it step by step:

Step 1: Set up synthetic division:

-4 | 1 0 -28 -48

Step 2: Bring down the coefficient of the highest degree term (the 1 in this case):

-4 | 1 0 -28 -48
____________________


Step 3: Multiply the divisor (-4) by the first term (1), then write the result under the second term (0):

-4 | 1 0 -28 -48
-4

Step 4: Add the result obtained in step 3 to the second term (-4 + 0 = -4), write it below, then repeat the process:

-4 | 1 0 -28 -48
-4
______________
1

Step 5: Multiply the divisor (-4) by the result obtained in step 4 (1), then write the result below the third term (-28):

-4 | 1 0 -28 -48
-4 16

Step 6: Add the result obtained in step 5 to the third term (-28 + 16 = -12), write it below, then repeat the process:

-4 | 1 0 -28 -48
-4 16
______________
1 -12

Step 7: Multiply the divisor (-4) by the result obtained in step 6 (-12), then write the result below the fourth term (-48):

-4 | 1 0 -28 -48
-4 16 48
__________________

Step 8: Add the result obtained in step 7 to the fourth term (-48 + 48 = 0), write it below:

-4 | 1 0 -28 -48
-4 16 48
_______________
1 -12 20 0

Step 9: Read the coefficients of the quotient from the bottom row (1, -12, 20), which represents the factored form of the polynomial.

Therefore, the factored form of the function f(x) = x^3 - 28x - 48 is:

f(x) = (x + 4)(x^2 - 12x + 20)

Note: You can also verify your result by multiplying back the factors to see if you recover the original polynomial.