F(x) = x⁵-4x⁴+3x³+2x²+x-6 dibagi (x²-2x+3)

We are tasked with dividing the function \( F(x) = x^5 - 4x^4 + 3x^3 + 2x^2 + x - 6 \) by \( x^2 - 2x + 3 \).

This division can be solved using **long division of polynomials**. I'll walk you through the steps:

### Step-by-Step Solution:

1. **Divide the leading term of the dividend** by the leading term of the divisor:
   \[
   \frac{x^5}{x^2} = x^3
   \]

2. **Multiply the divisor** \( x^2 - 2x + 3 \) by \( x^3 \):
   \[
   x^3(x^2 - 2x + 3) = x^5 - 2x^4 + 3x^3
   \]

3. **Subtract this from the original polynomial**:
   \[
   (x^5 - 4x^4 + 3x^3 + 2x^2 + x - 6) - (x^5 - 2x^4 + 3x^3) = -2x^4 + 2x^2 + x - 6
   \]

4. **Repeat the process** with the new polynomial:
   \[
   \frac{-2x^4}{x^2} = -2x^2
   \]

5. **Multiply the divisor** by \( -2x^2 \):
   \[
   -2x^2(x^2 - 2x + 3) = -2x^4 + 4x^3 - 6x^2
   \]

6. **Subtract**:
   \[
   (-2x^4 + 2x^2 + x - 6) - (-2x^4 + 4x^3 - 6x^2) = -4x^3 + 8x^2 + x - 6
   \]

7. **Continue the process**:
   \[
   \frac{-4x^3}{x^2} = -4x
   \]
   Multiply the divisor by \( -4x \):
   \[
   -4x(x^2 - 2x + 3) = -4x^3 + 8x^2 - 12x
   \]
   Subtract:
   \[
   (-4x^3 + 8x^2 + x - 6) - (-4x^3 + 8x^2 - 12x) = 13x - 6
   \]

8. **Final step**:
   \[
   \frac{13x}{x^2} = 13
   \]
   Multiply the divisor by 13:
   \[
   13(x^2 - 2x + 3) = 13x^2 - 26x + 39
   \]
   Subtract:
   \[
   (13x - 6) - (13x^2 - 26x + 39) = -13x^2 + 39x - 45
   \]

### Final Answer:
The result of the division is:
\[
F(x) \div (x^2 - 2x + 3) = x^3 - 2x^2 - 4x + 13 \text{ with a remainder of } -13x^2 + 39x - 45
\]

Would you like a detailed explanation of any specific step?

### Further Questions:
1. What are the steps to perform polynomial long division?
2. How do we verify the result of polynomial division?
3. What is synthetic division, and how does it differ from long division?
4. How is the remainder theorem used in polynomial division?
5. How would you find the quotient when dividing a cubic polynomial?

**Tip:** Always align the powers of \( x \) correctly when performing polynomial long division for accuracy.

Explore a detailed step-by-step solution for dividing the polynomial F(x) = x⁵-4x⁴+3x³+2x²+x-6 by x²-2x+3 using polynomial long division. This problem covers algebraic concepts, polynomial division, and the Polynomial Division Theorem, suitable for high school 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