(x4−5x3−8x2+13x−12)÷(x−6) Synthetic Division

Jasonbradley

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Jun 17, 2024

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Synthetic Division: (x⁴−5x³−8x² +13x−12) ÷ (x−6)

Synthetic division is a shorthand method for dividing polynomials, particularly when the divisor is a linear expression in the form (x - a). Let's break down how to use synthetic division to solve the problem (x⁴−5x³−8x² +13x−12) ÷ (x−6).

Step 1: Setting up the problem

1. Identify the coefficients: Write down the coefficients of the dividend polynomial, ensuring to include a coefficient of 0 for any missing terms in the descending order of powers:

   1  -5  -8  13  -12 
   

2. Identify the divisor: The divisor is (x - 6). Take the opposite of the constant term, which is 6.

Step 2: Performing the division

1. Bring down the first coefficient: Bring down the first coefficient, 1, below the line.

       6 | 1  -5  -8  13  -12 
           --------------------
               1 
   

2. Multiply and add: Multiply the number you just brought down (1) by the divisor (6) and place the product (6) under the next coefficient (-5). Add the two numbers together (-5 + 6 = 1) and write the result below the line.

       6 | 1  -5  -8  13  -12 
           --------------------
               1   1
   

3. Repeat: Continue this process, multiplying the number you just wrote below the line (1) by the divisor (6) and adding it to the next coefficient (-8).

       6 | 1  -5  -8  13  -12 
           --------------------
               1   1   2
   

4. Continue until you reach the last coefficient: Repeat the process until you reach the last coefficient.

       6 | 1  -5  -8  13  -12 
           --------------------
               1   1   2   25
   

5. The remainder: The last number below the line is the remainder.

       6 | 1  -5  -8  13  -12 
           --------------------
               1   1   2   25   98
   

Step 3: Interpreting the result

The numbers below the line represent the coefficients of the quotient polynomial, starting from the highest degree:

• Quotient: x³ + x² + 2x + 25
• Remainder: 98

Therefore, (x⁴−5x³−8x² +13x−12) ÷ (x−6) = x³ + x² + 2x + 25 + 98/(x-6)