solve this problem usin synthetic division 5x⁴ -2x² -3÷ x-1

To solve the problem \( 5x^4 - 2x^2 - 3 \) divided by \( x - 1 \) using synthetic division, follow these steps:

### Step 1: Set Up the Synthetic Division

The coefficients of the polynomial \( 5x^4 - 2x^2 - 3 \) are:  
- \( 5 \) (coefficient of \( x^4 \))
- \( 0 \) (coefficient of \( x^3 \), since there's no \( x^3 \) term)
- \( -2 \) (coefficient of \( x^2 \))
- \( 0 \) (coefficient of \( x \), since there's no \( x \) term)
- \( -3 \) (constant term)

The divisor \( x - 1 \) corresponds to \( x = 1 \).

So, we set up the synthetic division as follows:

```
1 |  5   0   -2   0   -3
```

### Step 2: Perform the Division

1. **Bring down** the first coefficient (which is 5):
   \[
   5
   \]
2. **Multiply** 1 by the number you just wrote down (5), and **write the result under the next coefficient**:
   \[
   1 \times 5 = 5
   \]
   Place 5 under the next coefficient (0):
   \[
   1 |  5   0   -2   0   -3
       |      5
   \]
3. **Add** the numbers in the second column:
   \[
   0 + 5 = 5
   \]
   Write the sum below the line:
   \[
   1 |  5   0   -2   0   -3
       |      5
       ----------------
          5
   \]
4. **Repeat** this process for the rest of the coefficients:
   - Multiply 1 by 5, add to -2:
     \[
     1 \times 5 = 5, \quad -2 + 5 = 3
     \]
   - Multiply 1 by 3, add to 0:
     \[
     1 \times 3 = 3, \quad 0 + 3 = 3
     \]
   - Multiply 1 by 3, add to -3:
     \[
     1 \times 3 = 3, \quad -3 + 3 = 0
     \]
  
   This leaves the synthetic division table looking like this:

   ```
   1 |  5   0   -2   0   -3
       |      5   5   3   3
       ----------------------
          5   5   3   3   0
   ```

### Step 3: Write the Result

The result from synthetic division is:

\[
5x^3 + 5x^2 + 3x + 3
\]

The remainder is \( 0 \).

So, the division of \( 5x^4 - 2x^2 - 3 \) by \( x - 1 \) results in:

\[
\boxed{5x^3 + 5x^2 + 3x + 3}
\]

Would you like further details or have any questions?

---

Here are 5 related questions for you to consider:

1. How would the result change if the divisor were \( x + 1 \) instead of \( x - 1 \)?
2. What would be the remainder if the synthetic division resulted in a non-zero number?
3. How does synthetic division compare to polynomial long division?
4. Can synthetic division be used if the divisor is not a linear polynomial (e.g., \( x^2 + 1 \))?
5. How can you verify the result of the division?

**Tip:** Always check your polynomial's coefficients for any missing terms and insert zeroes accordingly before performing synthetic division.

solve this problem using synthetic division 5x⁴ -2x² -3÷ x-1

Understand how to perform synthetic division for the polynomial 5x⁴ - 2x² - 3 divided by x - 1 with a step-by-step solution. This guide covers key concepts in algebra, including synthetic division and the remainder theorem, suitable for high school students.

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