Answer:
The zeros of the polynomial ( f(x) = x^3 + 5x^2 - 2x - 24 ) are 2, -3, and -4.
Explanation:
To find all zeros of the polynomial ( f(x) = x^3 + 5x^2 - 2x - 24 ) when given the factor ( (x-2) ), we can use synthetic division to divide the polynomial by ( (x-2) ).
The synthetic division process involves using the root (in this case, 2) as the divisor and performing the division to find the quotient and remainder.
The remainder should be zero if the given root is indeed a zero of the polynomial.
Performing the synthetic division:
2 │ 1 5 -2 -24
└ 2 14 24
-----------------
1 7 12 0
The result of the synthetic division is 1 7 12 0, which corresponds to the coefficients of the quotient polynomial.
This means that the quotient polynomial is ( x^2 + 7x + 12 ).
Now, we can factor the quotient polynomial ( x^2 + 7x + 12 ) by finding its roots:
[ x^2 + 7x + 12 = (x+3)(x+4) ]
Thus, the zeros of the polynomial ( f(x) = x^3 + 5x^2 - 2x - 24 ) are 2, -3, and -4.