To solve the expression \((2x^3 - 8x^2 - 12x + 5) \div (x - 3)\), we can perform polynomial long division.

### Steps:

1. **Divide the first term:**  
   Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\).  
   \[
   \frac{2x^3}{x} = 2x^2
   \]
   This gives the first term of the quotient.

2. **Multiply and subtract:**  
   Multiply \(2x^2\) by the entire divisor \((x - 3)\):  
   \[
   2x^2(x - 3) = 2x^3 - 6x^2
   \]
   Subtract this from the original polynomial:  
   \[
   (2x^3 - 8x^2) - (2x^3 - 6x^2) = -2x^2
   \]

3. **Repeat the process:**  
   Divide \(-2x^2\) by \(x\):  
   \[
   \frac{-2x^2}{x} = -2x
   \]
   Multiply \(-2x\) by \((x - 3)\):  
   \[
   -2x(x - 3) = -2x^2 + 6x
   \]
   Subtract:  
   \[
   (-2x^2 - 12x) - (-2x^2 + 6x) = -18x
   \]

4. **Continue dividing:**  
   Divide \(-18x\) by \(x\):  
   \[
   \frac{-18x}{x} = -18
   \]
   Multiply \(-18\) by \((x - 3)\):  
   \[
   -18(x - 3) = -18x + 54
   \]
   Subtract:  
   \[
   (-18x + 5) - (-18x + 54) = -49
   \]

### Final Answer:

The quotient is \(2x^2 - 2x - 18\), and the remainder is \(-49\).

So, the result of the division is:
\[
\boxed{2x^2 - 2x - 18 + \frac{-49}{x - 3}}
\]

Would you like further details or have any questions?

### Relative Questions:

1. How is polynomial long division different from numerical long division?
2. What are some common applications of polynomial division?
3. How can synthetic division be used in place of polynomial long division?
4. What is the significance of the remainder in polynomial division?
5. How do you find the roots of a polynomial after dividing by a linear factor?

### Tip:

When performing polynomial division, always check your work by multiplying the quotient by the divisor and adding the remainder. This should yield the original polynomial.

Learn how to perform polynomial long division step-by-step with (2x^3 - 8x^2 - 12x + 5) ÷ (x - 3). This problem involves dividing a polynomial by a linear factor, covering key concepts in algebra. 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