https://file.aiquickdraw.com/mathbot/file/1727069764208nawgqdu9.jpg

The problem shown in the image asks you to perform long division of two polynomials:

\[
\frac{x^5 - 6x^4 + 9x^3 - 20x + 24}{x^2 - 2}
\]

Let's work through this step by step using polynomial long division.

### Step 1: Set up the division
We are dividing \(x^5 - 6x^4 + 9x^3 - 20x + 24\) by \(x^2 - 2\).

### Step 2: Divide the first term of the dividend by the first term of the divisor.
\[
\frac{x^5}{x^2} = x^3
\]
So, the first term of the quotient is \(x^3\).

### Step 3: Multiply the divisor by \(x^3\).
\[
(x^2 - 2) \times x^3 = x^5 - 2x^3
\]

### Step 4: Subtract this from the original polynomial.
\[
(x^5 - 6x^4 + 9x^3 - 20x + 24) - (x^5 - 2x^3) = -6x^4 + 11x^3 - 20x + 24
\]

### Step 5: Repeat the process with the new polynomial.

1. Divide the first term: \(\frac{-6x^4}{x^2} = -6x^2\).
2. Multiply: \((x^2 - 2) \times (-6x^2) = -6x^4 + 12x^2\).
3. Subtract: \((-6x^4 + 11x^3 - 20x + 24) - (-6x^4 + 12x^2) = 11x^3 - 12x^2 - 20x + 24\).

### Step 6: Continue dividing.
1. Divide \(\frac{11x^3}{x^2} = 11x\).
2. Multiply: \((x^2 - 2) \times 11x = 11x^3 - 22x\).
3. Subtract: \((11x^3 - 12x^2 - 20x + 24) - (11x^3 - 22x) = -12x^2 + 2x + 24\).

### Step 7: Final steps.
1. Divide \(\frac{-12x^2}{x^2} = -12\).
2. Multiply: \((x^2 - 2) \times (-12) = -12x^2 + 24\).
3. Subtract: \((-12x^2 + 2x + 24) - (-12x^2 + 24) = 2x\).

### Final Answer:
The quotient is:
\[
x^3 - 6x^2 + 11x - 12
\]
with a remainder of \(2x\), so the complete answer is:
\[
x^3 - 6x^2 + 11x - 12 + \frac{2x}{x^2 - 2}
\]

Would you like any additional details or clarifications?

Here are 5 questions that build on this concept:
1. What is the significance of the remainder in polynomial division?
2. How can polynomial division be applied to factor polynomials?
3. How would synthetic division be used instead of long division for this problem?
4. What happens if the degree of the divisor is greater than the degree of the dividend?
5. How do you check if your polynomial division answer is correct?

**Tip**: When dividing polynomials, make sure to align terms by degree for easier subtraction during the long division steps.

Use long division to divide. Express any numbers as integers or simplified fractions. Divide (x^5 - 6x^4 + 9x^3 - 20x + 24) by (x^2 - 2).

Learn how to divide polynomials using long division with the expression (x^5 - 6x^4 + 9x^3 - 20x + 24) divided by (x^2 - 2). This step-by-step solution covers polynomial long division, showing all necessary steps for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation