To find the third zero and the values of a and b, we can use the fact that the sum of the zeros of a cubic polynomial is equal to the opposite of the coefficient of the squared term divided by the coefficient of the cubed term.
Given that two of the zeros are -1 and -2, let's find the third zero:
1. Let's assume the third zero is represented by the variable 'c.'
2. By the zero product property, we know that (x+1)(x+2)(x+c) = 0 represents the given cubic polynomial.
3. Expanding the above expression, we have (x^2+3x+2)(x+c) = 0.
4. Rearranging the terms, we get (x^3 + cx^2 + 3x^2 + 3cx + 2x + 2c) = 0.
5. Combining like terms, we have x^3 + (c+3)x^2 + (3c+2)x + 2c = 0.
Comparing this equation with the given cubic polynomial ax^3 + 3x^2 - bx - 6, we can equate the corresponding coefficients to find the values of a, b, and c.
From the equation, we can deduce the following:
Cubic Term: a = 1 (coefficient of x^3 term)
Squared Term: c+3 = 3 (coefficient of x^2 term)
Linear Term: 3c+2 = -b (coefficient of x term)
Constant Term: 2c = -6 (constant term)
Simplifying these equations, we find:
c = -3 (subtract 3 from both sides of c+3 = 3)
b = 1 (replace c with -3 in 3c+2 = -b)
2c = -6 (substitute c = -3 into 2c = -6)
So, the third zero is -3, and the values of a and b are 1 and -6, respectively.