(12x^4+20x^3-24x^2+20x+35)/(3x+5)

3 min read Jun 16, 2024
(12x^4+20x^3-24x^2+20x+35)/(3x+5)

Dividing Polynomials: A Step-by-Step Guide

This article will guide you through the process of dividing the polynomial 12x⁴ + 20x³ - 24x² + 20x + 35 by 3x + 5. We will utilize the long division method for this.

Setting Up the Long Division

  1. Arrange the terms: Write the dividend (12x⁴ + 20x³ - 24x² + 20x + 35) inside the long division symbol and the divisor (3x + 5) outside. Make sure the terms are arranged in descending order of their exponents.

         ________________________
    3x + 5 | 12x⁴ + 20x³ - 24x² + 20x + 35 
    

Performing the Division

  1. Divide the leading terms: Divide the leading term of the dividend (12x⁴) by the leading term of the divisor (3x). This gives us 4x³. Write this result above the dividend, aligned with the x³ term.

         4x³ ________________________
    3x + 5 | 12x⁴ + 20x³ - 24x² + 20x + 35 
    
  2. Multiply and subtract: Multiply the quotient (4x³) by the divisor (3x + 5) and write the result below the dividend. Then, subtract this product from the dividend.

         4x³ ________________________
    3x + 5 | 12x⁴ + 20x³ - 24x² + 20x + 35 
             -(12x⁴ + 20x³)
             -------------------
                     -24x² + 20x + 35
    
  3. Bring down the next term: Bring down the next term of the dividend (-24x²) to the bottom row.

         4x³ ________________________
    3x + 5 | 12x⁴ + 20x³ - 24x² + 20x + 35 
             -(12x⁴ + 20x³)
             -------------------
                     -24x² + 20x + 35
    
  4. Repeat the process: Repeat steps 2-4 until there are no more terms in the dividend to bring down.

         4x³ - 8x ________________________
    3x + 5 | 12x⁴ + 20x³ - 24x² + 20x + 35 
             -(12x⁴ + 20x³)
             -------------------
                     -24x² + 20x + 35
                     -(-24x² - 40x)
                     -------------------
                             60x + 35
    
         4x³ - 8x + 20 ________________________
    3x + 5 | 12x⁴ + 20x³ - 24x² + 20x + 35 
             -(12x⁴ + 20x³)
             -------------------
                     -24x² + 20x + 35
                     -(-24x² - 40x)
                     -------------------
                             60x + 35
                             -(60x + 100)
                             -------------------
                                     -65
    

The Result

Therefore, the quotient of (12x⁴ + 20x³ - 24x² + 20x + 35) divided by (3x + 5) is 4x³ - 8x + 20 with a remainder of -65. This can be expressed as:

12x⁴ + 20x³ - 24x² + 20x + 35 = (3x + 5)(4x³ - 8x + 20) - 65