(4𝑥^2− 2𝑥 + 5) ÷ (𝑥 − 1) use synthetic division

To solve \( \frac{4x^2 - 2x + 5}{x - 1} \) using synthetic division, follow these steps:

### Step 1: Set up the synthetic division.
For synthetic division, we focus on the divisor \(x - 1\), where the zero is \(x = 1\).

The dividend coefficients are \( 4x^2 - 2x + 5 \), so the coefficients are: \(4, -2, 5\).

### Step 2: Perform synthetic division.
- Write the coefficients of the dividend: \(4, -2, 5\).
- Write \(1\) (from \(x = 1\)) to the left.

\[
\begin{array}{r|rrr}
1 & 4 & -2 & 5 \\
  &   & 4  & 2  \\
\hline
  & 4 & 2  & 7  \\
\end{array}
\]

### Step 3: Process.
1. Bring down the \(4\) to the bottom row.
2. Multiply \(4\) by \(1\) (the divisor's root), getting \(4\), and add it to the next coefficient: \( -2 + 4 = 2 \).
3. Multiply \(2\) by \(1\), getting \(2\), and add it to the next coefficient: \(5 + 2 = 7\).

### Step 4: Interpret the result.
The quotient is \(4x + 2\), and the remainder is \(7\). So, the result of the division is:

\[
\frac{4x^2 - 2x + 5}{x - 1} = 4x + 2 + \frac{7}{x - 1}
\]

This is the final result using synthetic division.

Would you like more details or have any questions?

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Here are 5 related questions to explore this further:
1. How does synthetic division differ from long division in polynomials?
2. Can synthetic division be used when the divisor is not in the form \(x - c\)?
3. What happens when you perform synthetic division and the remainder is zero?
4. How would the process change if you were dividing a cubic polynomial?
5. What are the applications of synthetic division in solving algebraic problems?

**Tip:** When using synthetic division, always ensure the divisor is of the form \(x - c\) to apply it correctly.

Learn how to divide the polynomial (4x^2 - 2x + 5) by (x - 1) using synthetic division. This step-by-step guide explains the process of synthetic division and provides insights into key algebraic concepts like polynomials and the Remainder Theorem. Suitable for students in Grades 9-11.

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