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The problem in the image asks to factor the polynomial \( f(x) = 4x^3 + 19x^2 - 101x + 24 \) completely, given that \( -8 \) is a zero of \( f(x) \). This means that \( x + 8 \) is a factor of \( f(x) \). We can solve this by using synthetic division to divide the polynomial by \( x + 8 \).

### Step-by-Step Solution:

1. **Set up for Synthetic Division**:
   Since \( -8 \) is a root, divide \( f(x) \) by \( x + 8 \) using synthetic division.

2. **Coefficients** of \( f(x) \): 
   The polynomial \( f(x) = 4x^3 + 19x^2 - 101x + 24 \) has coefficients \( [4, 19, -101, 24] \).

3. **Perform Synthetic Division**:
   We use the root \( -8 \) to divide:

   \[
   \begin{array}{r|rrrr}
     -8 & 4 & 19 & -101 & 24 \\
        &   & -32 & 104  & -24 \\
     \hline
        & 4 & -13 & 3    & 0 \\
   \end{array}
   \]

   The quotient is \( 4x^2 - 13x + 3 \), and the remainder is 0, confirming that \( x + 8 \) is a factor.

4. **Factor the Quotient**:
   Now factor the quadratic \( 4x^2 - 13x + 3 \).
   
   We look for two numbers that multiply to \( 4 \times 3 = 12 \) and add to \( -13 \). These numbers are \( -12 \) and \( -1 \).

   So, rewrite the middle term:
   \[
   4x^2 - 12x - x + 3
   \]
   Factor by grouping:
   \[
   4x(x - 3) - 1(x - 3)
   \]
   Factor out \( (x - 3) \):
   \[
   (4x - 1)(x - 3)
   \]

5. **Final Factored Form**:
   Therefore, the fully factored form of \( f(x) \) is:
   \[
   f(x) = (x + 8)(4x - 1)(x - 3)
   \]

Let me know if you'd like further clarification or details on any step.

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### Follow-up Questions:
1. What other methods can you use to check if a factor is correct?
2. How do you perform synthetic division if the divisor is not linear?
3. What happens when the remainder from synthetic division is not zero?
4. How would the approach change if the leading coefficient wasn't 4?
5. Can you use this method to find all roots of any cubic polynomial?

### Tip:
Always check your factored form by multiplying the factors back together to confirm you get the original polynomial!

Factor f(x) = 4x^3 + 19x^2 - 101x + 24 into linear factors given that -8 is a zero of f(x).

Learn how to completely factor the cubic polynomial 4x^3 + 19x^2 - 101x + 24 into linear factors, given that -8 is a zero. This step-by-step solution includes synthetic division and quadratic factorization techniques, suitable for high school students.

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