Final answer:
The question asks to verify if -2 is a solution to the polynomial -x^3 + 14x^2 - 32x - 128 = 0 using synthetic division. Synthetic division involves using the coefficient of the polynomial and the value to be tested, following a step-by-step process to determine if the remainder is zero.
Step-by-step explanation:
The subject of this question is Mathematics, specifically regarding the use of synthetic division to find solutions to polynomial equations. The provided equation is a cubic polynomial, and the student is asked to use synthetic division to verify if -2 is a solution to the equation -x^3 + 14x^2 - 32x - 128 = 0.
Therefore, after performing synthetic division, the quotient is -x² + 16x - 64, and the remainder is -60.
The equation -x³ + 14x² - 32x - 128 = 0 factors as (-x - 2)(x² - 16x + 64) = 0.
Setting each factor equal to zero:
-x - 2 = 0
Solving for x:
-x = 2
x = -2 (which is the given root)
x² - 16x + 64 = 0
Factorizing the quadratic equation:
(x - 8)(x - 8) = 0
x - 8 = 0
x = 8
Hence, the solutions to the equation -x³ + 14x² - 32x - 128 = 0 are x = -2 and x = 8.