Determine all rational zeros of the polynomial P(x)=x3−x2−8x+12, and write the polynomial in factored form.
The leading coefficient of P is 1, so all the rational zeros are integers:
They are divisors of the constant term 12. Thus, the possible candidates are
±1,±2,±3,±4,±6,±12
Using Synthetic Division
We find that 1,3,4 and 6 are not zeros but that 2 is a zero and that P factors as
x3−x2−8x+12=(x−2)(x2+x−6)
We now factor x2+x−6 using trial and error, the factors are (x−2)(x+3), so
x3−x2−8x+12=(x−2)(x−2)(x+3)
Therefore, the zeros of P are 2 and −3.
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