Divide using: synthetic method

1.) 4x³- 6x²+5x-2/x-2
2.) 2x⁴-3x³+7x²/x+1

Let's solve these division problems using the synthetic division method.

### Problem 1: \( \frac{4x^3 - 6x^2 + 5x - 2}{x - 2} \)

1. **Set up the synthetic division:**
   - Write the coefficients of the polynomial: \( 4, -6, 5, -2 \).
   - Since we are dividing by \( x - 2 \), the root is \( 2 \).

2. **Perform the synthetic division:**

\[
\begin{array}{r|rrrr}
2 & 4 & -6 & 5 & -2 \\
  &   & 8  & 4 & 18  \\
\hline
  & 4 & 2  & 9 & 16 \\
\end{array}
\]

The steps:
- Bring down the first coefficient \(4\).
- Multiply \(4\) by \(2\) (gives \(8\)), and add to the next coefficient \(-6\) (gives \(2\)).
- Multiply \(2\) by \(2\) (gives \(4\)), and add to the next coefficient \(5\) (gives \(9\)).
- Multiply \(9\) by \(2\) (gives \(18\)), and add to the last coefficient \(-2\) (gives \(16\)).

3. **Interpret the result:**
   - The result is \(4x^2 + 2x + 9\) with a remainder of \(16\).
   - So, \( \frac{4x^3 - 6x^2 + 5x - 2}{x - 2} = 4x^2 + 2x + 9 + \frac{16}{x-2} \).

### Problem 2: \( \frac{2x^4 - 3x^3 + 7x^2}{x + 1} \)

1. **Set up the synthetic division:**
   - Write the coefficients of the polynomial: \( 2, -3, 7, 0, 0 \) (Note: the \(0\)s account for the missing terms \(x\) and the constant term).
   - Since we are dividing by \( x + 1 \), the root is \( -1 \).

2. **Perform the synthetic division:**

\[
\begin{array}{r|rrrrr}
-1 & 2 & -3 & 7 & 0 & 0 \\
   &   & -2 & 5 & -12 & 12 \\
\hline
   & 2 & -5 & 12 & -12 & 12 \\
\end{array}
\]

The steps:
- Bring down the first coefficient \(2\).
- Multiply \(2\) by \(-1\) (gives \(-2\)), and add to the next coefficient \(-3\) (gives \(-5\)).
- Multiply \(-5\) by \(-1\) (gives \(5\)), and add to the next coefficient \(7\) (gives \(12\)).
- Multiply \(12\) by \(-1\) (gives \(-12\)), and add to the next coefficient \(0\) (gives \(-12\)).
- Multiply \(-12\) by \(-1\) (gives \(12\)), and add to the last coefficient \(0\) (gives \(12\)).

3. **Interpret the result:**
   - The result is \(2x^3 - 5x^2 + 12x - 12\) with a remainder of \(12\).
   - So, \( \frac{2x^4 - 3x^3 + 7x^2}{x + 1} = 2x^3 - 5x^2 + 12x - 12 + \frac{12}{x+1} \).

Would you like further details on these calculations or have any questions? Here are some related problems to try:

1. Divide \( 3x^4 - 5x^3 + 2x^2 - x + 1 \) by \( x - 3 \) using synthetic division.
2. Divide \( x^5 - 2x^4 + 3x^3 - 4x^2 + 5x - 6 \) by \( x + 2 \).
3. Divide \( 6x^3 - 11x^2 + 7x - 4 \) by \( x - 1 \).
4. Divide \( 8x^4 + 0x^3 - 6x^2 + 9x - 12 \) by \( x + 3 \).
5. Divide \( 4x^2 - 7x + 10 \) by \( x - 5 \).

**Tip:** Synthetic division is a quick way to divide polynomials when the divisor is a linear binomial of the form \(x - c\).

Divide using: synthetic method
1.) 4x³- 6x²+5x-2/x-2
2.) 2x⁴-3x³+7x²/x+1

Explore how to divide polynomials using the synthetic division method. This guide covers the division of 4x³ - 6x² + 5x - 2 by x - 2 and 2x⁴ - 3x³ + 7x² by x + 1, with detailed steps and solutions. Ideal for students in Grades 10-12.

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