Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

 

x−2
, 4x^3−3x^2−8x−4

To determine whether \( x - 2 \) is a factor of \( 4x^3 - 3x^2 - 8x - 4 \) using synthetic division, we will follow these steps:

### Step 1: Set up the synthetic division
Since we are dividing by \( x - 2 \), the value to use in synthetic division is \( 2 \).

The coefficients of \( 4x^3 - 3x^2 - 8x - 4 \) are:  
\[ 4, -3, -8, -4 \]

So, the synthetic division setup looks like this:

\[
\begin{array}{r|rrrr}
2 & 4 & -3 & -8 & -4 \\
  &   & 8  & 10 & 4  \\
\hline
  & 4 & 5  & 2  & 0  \\
\end{array}
\]

### Step 2: Perform the synthetic division
- Bring down the first coefficient (4).
- Multiply \( 4 \) by \( 2 \) and place it under the next coefficient. \( 4 \times 2 = 8 \).
- Add \( -3 + 8 = 5 \).
- Multiply \( 5 \) by \( 2 \) and place it under the next coefficient. \( 5 \times 2 = 10 \).
- Add \( -8 + 10 = 2 \).
- Multiply \( 2 \) by \( 2 \) and place it under the next coefficient. \( 2 \times 2 = 4 \).
- Add \( -4 + 4 = 0 \).

### Step 3: Check the remainder
Since the remainder is 0, \( x - 2 \) is a factor of \( 4x^3 - 3x^2 - 8x - 4 \).

### Step 4: Write the factorization
The quotient from the synthetic division is \( 4x^2 + 5x + 2 \). Therefore, we can factor \( 4x^3 - 3x^2 - 8x - 4 \) as:
\[
(4x^3 - 3x^2 - 8x - 4) = (x - 2)(4x^2 + 5x + 2)
\]

Thus, the factorization is:
\[
4x^3 - 3x^2 - 8x - 4 = (x - 2)(4x^2 + 5x + 2)
\]

Let me know if you'd like more details or have any questions!

### Relative Questions:
1. How does synthetic division work step by step?
2. What is the difference between synthetic division and long division in algebra?
3. How can you verify if a factorization is correct?
4. What happens when the remainder in synthetic division is not zero?
5. Can synthetic division be used for divisors that are not linear (like \(x^2 + 3x + 1\))?

### Tip:
Always double-check your synthetic division result by multiplying the quotient with the divisor to see if you get the original polynomial back!

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

x−2, 4x^3−3x^2−8x−4

Determine if x - 2 is a factor of the polynomial 4x^3 - 3x^2 - 8x - 4 using synthetic division. This solution provides a detailed step-by-step explanation of the synthetic division process and verifies the factorization. Suitable for high school algebra students in Grades 9-12.

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