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

Jasonbradley

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·

Jun 16, 2024

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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

2. 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 
   

3. 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
   

4. 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
   

5. 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