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:

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!