(x^4+3x^3-4x^2+5x+3)/(x^2+x+4)

Jasonbradley

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Jun 17, 2024

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Polynomial Long Division: (x^4 + 3x^3 - 4x^2 + 5x + 3) / (x^2 + x + 4)

This article will guide you through the process of performing polynomial long division to find the quotient and remainder of the expression (x^4 + 3x^3 - 4x^2 + 5x + 3) / (x^2 + x + 4).

Understanding Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to the long division process we learn for numbers. It helps us determine the quotient and remainder when one polynomial is divided by another.

Steps for Polynomial Long Division

1. Set up the division. Write the dividend (x^4 + 3x^3 - 4x^2 + 5x + 3) inside the division symbol and the divisor (x^2 + x + 4) outside.

        ____________
   x^2+x+4 | x^4 + 3x^3 - 4x^2 + 5x + 3
   

2. Divide the leading terms. Divide the leading term of the dividend (x^4) by the leading term of the divisor (x^2). This gives us x^2. Write this above the division symbol.

        x^2        
   x^2+x+4 | x^4 + 3x^3 - 4x^2 + 5x + 3
   

3. Multiply the quotient by the divisor. Multiply the quotient (x^2) by the divisor (x^2 + x + 4). This gives us x^4 + x^3 + 4x^2.

        x^2        
   x^2+x+4 | x^4 + 3x^3 - 4x^2 + 5x + 3
           -(x^4 + x^3 + 4x^2)
   

4. Subtract. Subtract the result from the dividend.

        x^2        
   x^2+x+4 | x^4 + 3x^3 - 4x^2 + 5x + 3
           -(x^4 + x^3 + 4x^2)
           ------------------
                  2x^3 - 8x^2 + 5x 
   

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

        x^2        
   x^2+x+4 | x^4 + 3x^3 - 4x^2 + 5x + 3
           -(x^4 + x^3 + 4x^2)
           ------------------
                  2x^3 - 8x^2 + 5x + 3
   

6. Repeat steps 2-5. Repeat the process from step 2, dividing the new leading term (2x^3) by the divisor's leading term (x^2) to get 2x. Multiply 2x by the divisor, subtract the result, and bring down the next term (3).

        x^2 + 2x     
   x^2+x+4 | x^4 + 3x^3 - 4x^2 + 5x + 3
           -(x^4 + x^3 + 4x^2)
           ------------------
                  2x^3 - 8x^2 + 5x + 3
                  -(2x^3 + 2x^2 + 8x)
                  ------------------
                          -10x^2 - 3x + 3
   

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

        x^2 + 2x - 10
   x^2+x+4 | x^4 + 3x^3 - 4x^2 + 5x + 3
           -(x^4 + x^3 + 4x^2)
           ------------------
                  2x^3 - 8x^2 + 5x + 3
                  -(2x^3 + 2x^2 + 8x)
                  ------------------
                          -10x^2 - 3x + 3
                          -(-10x^2 - 10x - 40)
                          --------------------
                                 7x + 43
   

Solution

Therefore, the quotient of (x^4 + 3x^3 - 4x^2 + 5x + 3) / (x^2 + x + 4) is x^2 + 2x - 10 and the remainder is 7x + 43. This can be expressed as:

(x^4 + 3x^3 - 4x^2 + 5x + 3) / (x^2 + x + 4) = x^2 + 2x - 10 + (7x + 43) / (x^2 + x + 4)