Find the quotient and remainder using synthetic division. \[ \frac{x^{3}-5 x+22}{x+5} \] quotient remainder

To perform synthetic division for dividing x³ - 5x + 22 by x + 5, follow these steps: 1. Write the polynomial in standard form, including any missing terms. For x³ - 5x + 22, note that there is no x² term, so we use a coefficient of 0 for x²:   Coefficients: 1 (x³), 0 (x²), -5 (x), 22 (constant). 2. Set up synthetic division with the zero of the divisor. Since the divisor is x + 5, the zero is -5.     -5 | 1    0     -5    22        |   -5    25     -100     -------------------------        1   -5   20    -78 Steps: • Bring down the first coefficient (1). • Multiply 1 by -5, giving -5. Write this under the second coefficient (0). • Add 0 and -5 to get -5. • Multiply -5 by -5 to get 25. Write this under the third coefficient (-5). • Add -5 and 25 to get 20. • Multiply 20 by -5 to get -100. Write this under the constant term (22). • Add 22 and -100 to get -78. The bottom row gives the coefficients of the quotient and the remainder: • Quotient: x² - 5x + 20 • Remainder: -78 Therefore, the quotient is x² - 5x + 20 and the remainder is -78.