Use synthetic division to find the quotient and remainder when

x Superscript 5 Baseline minus x cubed plus xx5−x3+x

is divided by

x plus 1x+1.

Question content area bottom

Part 1

​Quotient:

enter your response here

​Remainder:

enter your response here

To use synthetic division to divide \(x^5 - x^3 + x\) by \(x + 1\), follow these steps:

1. **Set up the synthetic division:**
   - First, rewrite the polynomial \(x^5 - x^3 + x\) with all terms present:
     \[
     x^5 + 0x^4 - x^3 + 0x^2 + x + 0
     \]
   - The divisor is \(x + 1\), so we use \( -1 \) (the opposite of \(+1\)) as the value for synthetic division.
   
2. **Synthetic division table:**

   Set up the coefficients: \([1, 0, -1, 0, 1, 0]\), and use \( -1 \) for the division:

   \[
   \begin{array}{r|rrrrrr}
   -1 & 1 & 0 & -1 & 0 & 1 & 0 \\
     &   & -1 & 1  & 0  & 0  & -1 \\
   \hline
     & 1 & -1 & 0  & 0  & 1  & -1 \\
   \end{array}
   \]

3. **Interpret the result:**

   The bottom row represents the coefficients of the quotient and the remainder:
   - The quotient is \(1x^4 - 1x^3 + 0x^2 + 0x + 1\), or simply \(x^4 - x^3 + 1\).
   - The remainder is \(-1\).

Thus:

- **Quotient:** \(x^4 - x^3 + 1\)
- **Remainder:** \(-1\)

Would you like more details on synthetic division or further clarification?

Here are some related questions to deepen your understanding:
1. How does synthetic division differ from long division in polynomials?
2. How would the quotient change if the divisor was \(x - 1\) instead?
3. What happens if you divide \(x^5 - x^3 + x\) by \(x - 2\)?
4. Can synthetic division be used for divisors of degree higher than 1?
5. How can you verify the quotient and remainder found in synthetic division?

**Tip:** Always ensure that your polynomial is written in standard form (descending powers of \(x\)) and includes placeholders for any missing terms before using synthetic division.

Use synthetic division to find the quotient and remainder when x^5 - x^3 + x is divided by x + 1.

Learn how to use synthetic division to find the quotient and remainder when dividing the polynomial x^5 - x^3 + x by x + 1. This step-by-step solution includes key algebraic concepts like synthetic division and polynomial representation. Suitable for students in 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