Question

Use synthetic division to solve and show your work. (x³-2x²-14x+24)-:(x-4)

Answer 1

Using synthetic division, the polynomial x³ - 2x² - 14x + 24 divided by x - 4 results in a quotient of x² + 2x - 6 with a remainder of 0, indicating that x - 4 is a factor of the polynomial.

Step-by-step explanation:

Synthetic Division: Polynomial Division

To use synthetic division to divide the polynomial x³ - 2x² - 14x + 24 by x - 4, we follow these steps:

1. Write down the coefficients of the polynomial: 1, -2, -14, 24.
2. Write the zero of the divisor, x - 4, which is 4, to the left of a vertical bar.
3. To the right of the bar, bring down the leading coefficient (1).
4. Multiply the zero by the number just written to the right of the bar (4 × 1 = 4) and write the result below the next coefficient (-2).
5. Add the numbers in the second column (-2 + 4 = 2).
6. Continue this process of multiplying and adding down the line.
7. Write down the result, which represents the coefficients of the quotient polynomial.

The complete synthetic division process will look like this:

4 | 1 -2 -14 24
| 4 8 -24
| 1 2 -6 0

The remainder is 0, indicating x - 4 is a factor of the polynomial. The result of the division is the polynomial x² + 2x - 6. To check if the solution is correct, you can multiply the quotient by the divisor and add the remainder (if any) to see if you get the original polynomial back.