What are the quotient and remainder when 5 x Superscript 4 Baseline minus x squared is divided by x cubed minus x squared plus 2​?
Question content area bottom
Part 1
The quotient is 
  
enter your response here and the remainder is 
  
enter your response here.

To find the quotient and remainder when dividing \( 5x^4 - x^2 \) by \( x^3 - x^2 + 2 \), let's use polynomial long division. 

### Step 1: Set Up the Division
We want to divide \( 5x^4 - x^2 \) (dividend) by \( x^3 - x^2 + 2 \) (divisor). First, we should write each polynomial with all missing terms filled in:

- **Dividend:** \( 5x^4 + 0x^3 - x^2 + 0x + 0 \)
- **Divisor:** \( x^3 - x^2 + 0x + 2 \)

### Step 2: Perform Polynomial Long Division
1. **Divide the leading term of the dividend by the leading term of the divisor:**

\[
\frac{5x^4}{x^3} = 5x
\]

2. **Multiply the entire divisor by \( 5x \) and subtract from the dividend:**

\[
5x(x^3 - x^2 + 2) = 5x^4 - 5x^3 + 10x
\]

Subtract from the dividend:

\[
(5x^4 + 0x^3 - x^2 + 0x + 0) - (5x^4 - 5x^3 + 10x) = 5x^3 - x^2 - 10x
\]

3. **Repeat the process using the new dividend \( 5x^3 - x^2 - 10x \):**

\[
\frac{5x^3}{x^3} = 5
\]

Multiply:

\[
5(x^3 - x^2 + 2) = 5x^3 - 5x^2 + 10
\]

Subtract:

\[
(5x^3 - x^2 - 10x) - (5x^3 - 5x^2 + 10) = 4x^2 - 10x - 10
\]

### Step 3: The Result
After completing the division, we get the quotient and remainder:

- **Quotient:** \( 5x + 5 \)
- **Remainder:** \( 4x^2 - 10x - 10 \)

So, the quotient is \( 5x + 5 \) and the remainder is \( 4x^2 - 10x - 10 \).

Would you like me to show the steps in more detail? Here are some related questions for practice:

1. What is the quotient when \( 6x^5 - 3x^3 + x \) is divided by \( x^2 - 2x + 3 \)?
2. How can you verify the quotient and remainder obtained from polynomial division?
3. What are the steps to use synthetic division for dividing a polynomial?
4. How does the degree of the remainder compare to the divisor in polynomial division?
5. What happens if the divisor has a higher degree than the dividend?

**Tip:** Always arrange the polynomials in descending order and fill in any missing terms before starting division.

What are the quotient and remainder when 5x^4 - x^2 is divided by x^3 - x^2 + 2?

Discover how to use polynomial long division to find the quotient and remainder when dividing 5x^4 - x^2 by x^3 - x^2 + 2. This problem demonstrates key concepts in algebra, including polynomial division and the division algorithm. 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