Answer:
{2, 2, -6}
Step-by-step explanation:
synthetic division is one of the easier ways of determining whether or not a given number is a root of a polynomial. Here we're told that -6 is a root. Let's go through the steps of synthetic division: keep in mind that if there is no remainder (that is, the remainder is zero), then the divisor (such as -6) is a zero of the polynomial.
The coefficients of f(x)=x^3+2x^2-20x+24 are {1, 2, -20, 24}.
Setting up synthetic div.:
-6 ) 1 2 -20 24
-6 24 -24
----------------------------
1 -4 4 0
Here the remainder is zero (0), so it is safe to assume that -6 is a zero of the original polynomial.
We can continue with synth. div. to determine the remaining zeros of the original function f(x)=x^3+2x^2-20x+24. The coefficients {1, -4, 4} represent the quadratic x^2 - 4x + 4. Let's check whether the factor 2 of 4 is indeed a zero:
2 ) 1 -4 4
2 -4
-----------------
1 -2 0
Here the remainder is zero. Thus, 2 is a zero of f(x)=x^3+2x^2-20x+24. Notice that the remaining coefficients are {1, -2}; they represent x - 2 = 0, so the third root is 2.
Thus, the zeros of f(x)=x^3+2x^2-20x+24 are {2, 2, -6}
