(4x4+5x−4)÷(x2−3x−2)

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (4x⁴ + 5x - 4) ÷ (x² - 3x - 2)

This article will guide you through the process of simplifying the expression (4x⁴ + 5x - 4) ÷ (x² - 3x - 2).

Understanding the Problem

We are dealing with a division problem involving polynomials. To simplify this, we'll utilize the long division method.

Long Division Steps

1. Set up the problem: Arrange the polynomials in descending order of their exponents, filling in any missing terms with coefficients of 0.

       ____________
   x² - 3x - 2 | 4x⁴ + 0x³ + 0x² + 5x - 4 
   

2. Divide the leading terms: Divide the leading term of the dividend (4x⁴) by the leading term of the divisor (x²), which gives 4x². Write this term above the line in the quotient.

       4x² ________
   x² - 3x - 2 | 4x⁴ + 0x³ + 0x² + 5x - 4 
   

3. Multiply the divisor by the quotient term: Multiply (x² - 3x - 2) by 4x².

       4x² ________
   x² - 3x - 2 | 4x⁴ + 0x³ + 0x² + 5x - 4 
                  4x⁴ - 12x³ - 8x²
   

4. Subtract: Subtract the result from the dividend.

       4x² ________
   x² - 3x - 2 | 4x⁴ + 0x³ + 0x² + 5x - 4 
                  4x⁴ - 12x³ - 8x²
                  -----------------
                         12x³ + 8x² + 5x 
   

5. Bring down the next term: Bring down the next term of the dividend (+ 5x).

       4x² ________
   x² - 3x - 2 | 4x⁴ + 0x³ + 0x² + 5x - 4 
                  4x⁴ - 12x³ - 8x²
                  -----------------
                         12x³ + 8x² + 5x 
   

6. Repeat steps 2-5: Divide the new leading term (12x³) by the divisor's leading term (x²), which gives 12x. Write this in the quotient, multiply, subtract, and bring down the next term.

       4x² + 12x _______
   x² - 3x - 2 | 4x⁴ + 0x³ + 0x² + 5x - 4 
                  4x⁴ - 12x³ - 8x²
                  -----------------
                         12x³ + 8x² + 5x 
                         12x³ - 36x² - 24x
                         -----------------
                                 44x² + 29x
   

7. Continue the process: Repeat steps 2-5 until the degree of the remaining polynomial is less than the degree of the divisor.

       4x² + 12x + 44 _______
   x² - 3x - 2 | 4x⁴ + 0x³ + 0x² + 5x - 4 
                  4x⁴ - 12x³ - 8x²
                  -----------------
                         12x³ + 8x² + 5x 
                         12x³ - 36x² - 24x
                         -----------------
                                 44x² + 29x - 4
                                 44x² - 132x - 88
                                 -----------------
                                         161x + 84 
   

Final Result

The simplified form of the expression is:

4x² + 12x + 44 + (161x + 84) / (x² - 3x - 2)

This represents the quotient (4x² + 12x + 44) and the remainder (161x + 84) over the original divisor.