Given x − 6 is a factor of f(x) = x3 + 5x2 − 48x − 108, determine the othertwo linear factors of f(x).

To find the other two linear factors of the polynomial f(x) = x³ + 5x² - 48x - 108, given that x - 6 is a factor, we can use polynomial division or synthetic division. Here, we'll use synthetic division.

Step 1: Set up the synthetic division. Write the coefficients of the polynomial above the division bar and the zero of the given factor (which is 6 because x - 6 = 0 when x = 6) to the left of the division bar.

        6 | 1   5  -48  -108
          |     6   66   108
          |_________________
            1  11   18    0

Step 2: Perform the synthetic division. Bring down the leading coefficient (1). Multiply the number you just wrote down by the zero of the factor (6), and write the result under the next coefficient (5). Add the numbers in this column and write the result below the line. Repeat this process for the rest of the coefficients.

Step 3: Interpret the result. The numbers on the bottom line represent the coefficients of the quotient polynomial, which is the result of the division. The degree of this polynomial is one less than the degree of the original polynomial. So, in this case, the quotient polynomial is x² + 11x + 18.

Step 4: Factor the quotient polynomial. The factors of 18 that add up to 11 are 2 and 9. So, x² + 11x + 18 = (x + 2)(x + 9).

Therefore, the three linear factors of the original polynomial are x - 6, x + 2, and x + 9.