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

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

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

Bring down the first coefficient: Bring down the first coefficient, 1, below the line.
6  1 5 8 13 12  1

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

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

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

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/(x6)