Problem 22 - College Algebra

Question

Divide using synthetic division. $$\left(5 x^{3}-6 x^{2}+3 x+11\right) \div(x-2)$$

Step-by-Step Solution

Answer

The result of the division is \(5x^2 - 16x + 32 + \frac{75}{x-2}\).

Step 1: Set Up the Synthetic Division

Write the coefficients of the polynomial to be divided at the top of the synthetic division table. For the polynomial \(5x^3 - 6x^2 + 3x + 11\), the coefficients are 5, -6, 3, and 11. Then, to the left of the bar, write the value that makes the divisor equal to zero; in this case, the expression is \(x - 2\), so the value is 2.

Step 2: Execute the Synthetic Division

Drop the first coefficient (5) down to the bottom row. Multiply this coefficient by the value from the divisor (2), then add the product to the next coefficient in the top row (-6) to get the next coefficient for the bottom row. Repeat this process until all coefficients have been used.

Step 3: Interpret the Results

The bottom row of coefficients represent the coefficients of the quotient. The degree of the quotient is always one less than the degree of the dividend. The last number in the bottom row is the remainder. Thus, the quotient polynomial is \(5x^2 - 16x + 32\), and the remainder is 75.