Given one zero of the polynomial function, find the other zeros. f(x)=x^3+2x^2-20x+24; -6

{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}

8

step-by-step explanation:

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step-by-step explanation: