Divide using synthetic division.
left parenthesis 4 x Superscript 5 Baseline minus 3 x cubed plus 2 x squared minus 3 x plus 5 right parenthesis divided by left parenthesis x minus 2 right parenthesis  

We can perform **synthetic division** on the polynomial \( 4x^5 - 3x^3 + 2x^2 - 3x + 5 \) by dividing it by \( x - 2 \).

### Step-by-Step Process:

1. **Write down the coefficients** of the polynomial in descending order of the powers of \( x \):
   - For \( 4x^5 - 3x^3 + 2x^2 - 3x + 5 \), the coefficients are:  
     \( 4, 0, -3, 2, -3, 5 \)  
     (Note: We need to include \( 0 \) for the missing \( x^4 \) term.)

2. **Set up the synthetic division table**:
   - Write the root of \( x - 2 = 0 \), which is \( 2 \), on the left side of the synthetic division table.
   - Write the coefficients in a row on the right:  
     \( 4, 0, -3, 2, -3, 5 \).

3. **Begin the synthetic division**:
   - Bring down the first coefficient \( 4 \) as is.
   - Multiply \( 4 \) by \( 2 \) (the divisor), and write the result under the next coefficient.
   - Add the numbers in this column.
   - Repeat this process for each column.

|     2     |  4    |   0   |  -3  |   2  |  -3  |   5  |
|-----------|-------|-------|------|------|------|------|
|           |       |   8   |  16  |  26  |  56  | 106  |
|    **↓**  |   4   |   8   |  13  |  18  |  53  | 109  |

4. **Interpret the result**:
   - The numbers in the bottom row represent the coefficients of the quotient and the remainder. So the quotient is:
     \[
     4x^4 + 8x^3 + 13x^2 + 18x + 53
     \]
   - The remainder is \( 109 \).

### Final Answer:

\[
\frac{4x^5 - 3x^3 + 2x^2 - 3x + 5}{x - 2} = 4x^4 + 8x^3 + 13x^2 + 18x + 53 + \frac{109}{x - 2}
\]

Would you like more details or have any questions?

---

Here are some related questions to expand your understanding:
1. How would the synthetic division change if the divisor was \( x + 2 \)?
2. Can synthetic division be applied when dividing by quadratic expressions?
3. What is the significance of the remainder in synthetic division?
4. How does synthetic division relate to polynomial long division?
5. Can synthetic division be used to factor polynomials?

**Tip:** Always check for any missing powers of \( x \) when setting up synthetic division.

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

Learn how to divide the polynomial 4x^5 - 3x^3 + 2x^2 - 3x + 5 using synthetic division with a detailed step-by-step solution. This problem involves algebraic concepts like polynomial division and is suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation