(x^3-8x^2+19x-15)/(x-3)

Jasonbradley

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·

Jun 17, 2024

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Dividing Polynomials: (x^3 - 8x^2 + 19x - 15) / (x - 3)

This article will demonstrate how to divide the polynomial (x^3 - 8x^2 + 19x - 15) by the binomial (x - 3) using polynomial long division.

Polynomial Long Division

1. Set up the division: Write the dividend (x^3 - 8x^2 + 19x - 15) inside the division symbol and the divisor (x - 3) outside.

        ___________
   x - 3 | x^3 - 8x^2 + 19x - 15 
   

2. Divide the leading terms: Divide the leading term of the dividend (x^3) by the leading term of the divisor (x). This gives us x^2, which we write above the division symbol.

        x^2        
   x - 3 | x^3 - 8x^2 + 19x - 15 
   

3. Multiply the divisor by the result: Multiply the divisor (x - 3) by the result we just obtained (x^2). This gives us x^3 - 3x^2. Write this product below the dividend, aligning the terms by their exponents.

        x^2        
   x - 3 | x^3 - 8x^2 + 19x - 15 
           x^3 - 3x^2
   

4. Subtract: Subtract the product from the dividend. Remember to change the signs of the terms being subtracted.

        x^2        
   x - 3 | x^3 - 8x^2 + 19x - 15 
           x^3 - 3x^2
           ---------
               -5x^2 
   

5. Bring down the next term: Bring down the next term of the dividend (+19x).

        x^2        
   x - 3 | x^3 - 8x^2 + 19x - 15 
           x^3 - 3x^2
           ---------
               -5x^2 + 19x 
   

6. Repeat steps 2-5: Divide the new leading term (-5x^2) by the leading term of the divisor (x). This gives us -5x. Write this above the division symbol.

        x^2 - 5x    
   x - 3 | x^3 - 8x^2 + 19x - 15 
           x^3 - 3x^2
           ---------
               -5x^2 + 19x 
               -5x^2 + 15x
   

7. Subtract: Subtract the product (-5x^2 + 15x) from the previous result.

        x^2 - 5x    
   x - 3 | x^3 - 8x^2 + 19x - 15 
           x^3 - 3x^2
           ---------
               -5x^2 + 19x 
               -5x^2 + 15x
               ---------
                      4x 
   

8. Bring down the next term: Bring down the last term of the dividend (-15).

        x^2 - 5x    
   x - 3 | x^3 - 8x^2 + 19x - 15 
           x^3 - 3x^2
           ---------
               -5x^2 + 19x 
               -5x^2 + 15x
               ---------
                      4x - 15 
   

9. Repeat steps 2-5: Divide the new leading term (4x) by the leading term of the divisor (x). This gives us +4. Write this above the division symbol.

        x^2 - 5x + 4
   x - 3 | x^3 - 8x^2 + 19x - 15 
           x^3 - 3x^2
           ---------
               -5x^2 + 19x 
               -5x^2 + 15x
               ---------
                      4x - 15 
                      4x - 12
   

10. Subtract: Subtract the product (4x - 12) from the previous result.

     x^2 - 5x + 4
x - 3 | x^3 - 8x^2 + 19x - 15 
        x^3 - 3x^2
        ---------
            -5x^2 + 19x 
            -5x^2 + 15x
            ---------
                   4x - 15 
                   4x - 12
                   ---------
                        -3 

Result

The result of the division is: x^2 - 5x + 4 with a remainder of -3. This can also be expressed as:

(x^3 - 8x^2 + 19x - 15) / (x - 3) = x^2 - 5x + 4 - 3/(x-3)

This means that the polynomial (x^3 - 8x^2 + 19x - 15) is equal to (x - 3) multiplied by (x^2 - 5x + 4) minus 3.