Dividing polynomials is a process similar to long division with numbers, but instead, you're dividing one polynomial by another. The goal is to express the quotient (result) and possibly a remainder, if there is one.

Here's a general step-by-step method for dividing polynomials:

### Example:
Divide \( 2x^3 + 3x^2 - 5x + 6 \) by \( x - 2 \).

### Steps:

1. **Divide the leading term**: Divide the first term of the dividend (the polynomial being divided) by the first term of the divisor (the polynomial you're dividing by). In this case:
   \[
   \frac{2x^3}{x} = 2x^2
   \]
   So, the first term of the quotient is \( 2x^2 \).

2. **Multiply and subtract**: Multiply the entire divisor by the result from step 1 (the quotient term) and subtract it from the dividend. 
   Multiply \( (x - 2) \) by \( 2x^2 \):
   \[
   2x^2(x - 2) = 2x^3 - 4x^2
   \]
   Subtract this from the original polynomial:
   \[
   (2x^3 + 3x^2 - 5x + 6) - (2x^3 - 4x^2) = 7x^2 - 5x + 6
   \]

3. **Repeat the process**: Now, divide the leading term of the new polynomial \( 7x^2 \) by the leading term of the divisor \( x \):
   \[
   \frac{7x^2}{x} = 7x
   \]
   Multiply \( (x - 2) \) by \( 7x \):
   \[
   7x(x - 2) = 7x^2 - 14x
   \]
   Subtract this from the current polynomial:
   \[
   (7x^2 - 5x + 6) - (7x^2 - 14x) = 9x + 6
   \]

4. **Continue until you can’t divide further**: Divide \( 9x \) by \( x \):
   \[
   \frac{9x}{x} = 9
   \]
   Multiply \( (x - 2) \) by \( 9 \):
   \[
   9(x - 2) = 9x - 18
   \]
   Subtract this from the polynomial:
   \[
   (9x + 6) - (9x - 18) = 24
   \]

5. **Write the result**: The quotient is \( 2x^2 + 7x + 9 \) and the remainder is \( 24 \). So, the result of the division is:
   \[
   \frac{2x^3 + 3x^2 - 5x + 6}{x - 2} = 2x^2 + 7x + 9 + \frac{24}{x - 2}
   \]

This is how you divide polynomials. The quotient is \( 2x^2 + 7x + 9 \), and the remainder is 24.

Would you like to dive into more details or try another example?

### Related Questions:
1. How do you perform synthetic division with polynomials?
2. How do you divide polynomials when the divisor has more than one term?
3. Can polynomials always be divided exactly, or are there cases with no remainder?
4. How does polynomial long division relate to factoring polynomials?
5. What is the Remainder Theorem, and how does it connect to polynomial division?

### Tip:
Always arrange the terms of the polynomials in descending order of the exponents before starting the division process.

Learn how to divide polynomials using long division with a detailed example: dividing 2x^3 + 3x^2 - 5x + 6 by x - 2. This guide covers key algebraic concepts, including quotient and remainder, and the application of the Remainder Theorem. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation