(x^3-4x^2-17x+6)/(x-3)

Jasonbradley

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Jun 17, 2024

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Dividing Polynomials: (x^3 - 4x^2 - 17x + 6) / (x - 3)

This article will guide you through the process of dividing the polynomial (x^3 - 4x^2 - 17x + 6) by the binomial (x - 3). We will use the long division method to achieve this.

Long Division Method

Step 1: Set up the division problem.

Write the problem as a long division, with the dividend (x^3 - 4x^2 - 17x + 6) under the division symbol and the divisor (x - 3) outside the symbol.

         ________
x - 3 | x^3 - 4x^2 - 17x + 6

Step 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. Write x^2 above the x^2 term in the dividend.

         x^2       
x - 3 | x^3 - 4x^2 - 17x + 6

Step 3: Multiply the quotient by the divisor.

Multiply the quotient (x^2) by the divisor (x - 3). This gives us x^3 - 3x^2. Write this result below the dividend, aligning like terms.

         x^2       
x - 3 | x^3 - 4x^2 - 17x + 6
         x^3 - 3x^2 

Step 4: Subtract.

Subtract the result from the previous step from the dividend. This will leave us with -x^2 - 17x.

         x^2       
x - 3 | x^3 - 4x^2 - 17x + 6
         x^3 - 3x^2 
         ---------
              -x^2 - 17x

Step 5: Bring down the next term.

Bring down the next term from the dividend (-17x) next to the result of the subtraction.

         x^2       
x - 3 | x^3 - 4x^2 - 17x + 6
         x^3 - 3x^2 
         ---------
              -x^2 - 17x + 6 

Step 6: Repeat steps 2-5.

Repeat the steps above, dividing the new leading term (-x^2) by the leading term of the divisor (x). This gives us -x. Write -x above the -17x term in the dividend.

         x^2 - x     
x - 3 | x^3 - 4x^2 - 17x + 6
         x^3 - 3x^2 
         ---------
              -x^2 - 17x + 6 
              -x^2 + 3x 

Subtract, bring down the next term, and repeat the process until there are no more terms to bring down.

         x^2 - x - 14
x - 3 | x^3 - 4x^2 - 17x + 6
         x^3 - 3x^2 
         ---------
              -x^2 - 17x + 6 
              -x^2 + 3x 
              ---------
                    -20x + 6
                    -20x + 60
                    ---------
                         -54

Step 7: Interpret the result.

The final result is the quotient (x^2 - x - 14) and a remainder of -54.

Therefore, we can express the division as:

(x^3 - 4x^2 - 17x + 6) / (x - 3) = x^2 - x - 14 - 54/(x - 3)