Divide using synthetic division.
left parenthesis 4 x cubed minus 4 x squared plus 5 x minus 5 right parenthesis divided by left parenthesis x minus 2 right parenthesis

To perform synthetic division, we'll divide the polynomial \( 4x^3 - 4x^2 + 5x - 5 \) by \( x - 2 \).

Here’s the step-by-step process:

### Step 1: Set up the synthetic division
- The divisor \( x - 2 \) gives a root of \( 2 \). We’ll use this in synthetic division.
- Write down the coefficients of the polynomial \( 4x^3 - 4x^2 + 5x - 5 \), which are \( 4, -4, 5, -5 \).

\[
\begin{array}{r|rrrr}
2 & 4 & -4 & 5 & -5 \\
\end{array}
\]

### Step 2: Perform synthetic division
1. Bring down the first coefficient, which is 4.
2. Multiply 4 by 2 (the root from the divisor), which gives \( 8 \). Write this under the next coefficient (-4).
3. Add the next column: \( -4 + 8 = 4 \).
4. Multiply 4 by 2, which gives \( 8 \). Write this under the next coefficient (5).
5. Add the next column: \( 5 + 8 = 13 \).
6. Multiply 13 by 2, which gives \( 26 \). Write this under the next coefficient (-5).
7. Add the next column: \( -5 + 26 = 21 \).

\[
\begin{array}{r|rrrr}
2 & 4 & -4 & 5 & -5 \\
  &   & 8  & 8  & 26 \\
\hline
  & 4 & 4  & 13 & 21 \\
\end{array}
\]

### Step 3: Write the result
The numbers on the bottom row \( 4, 4, 13 \) are the coefficients of the quotient, and the last number (21) is the remainder.

So the quotient is:

\[
4x^2 + 4x + 13
\]

And the remainder is \( 21 \), so the final result is:

\[
4x^2 + 4x + 13 + \frac{21}{x - 2}
\]

Would you like more details on any part of this solution?

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Here are five related questions to explore:
1. How does synthetic division differ from polynomial long division?
2. Can synthetic division be used when the divisor is not linear, like \( x^2 - 3 \)?
3. What happens if the divisor has a different root, like \( x + 1 \)?
4. How would you use synthetic division to find roots of a polynomial?
5. What is the significance of the remainder in synthetic division?

**Tip:** Always check if synthetic division is possible; it only works when the divisor is of the form \( x - c \).

Divide using synthetic division: (4x^3 - 4x^2 + 5x - 5) divided by (x - 2)

Learn how to perform synthetic division on the polynomial 4x^3 - 4x^2 + 5x - 5 divided by x - 2 with a step-by-step solution. This problem covers essential algebraic techniques including synthetic division and the remainder theorem, suitable for Grades 9-12.

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