Determine all rational zeros of the polynomial P(x)=x4−x3−23x2−3x+90, 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 90. Thus, the possible candidates are
±1,±2,±3,±5,±6,±9,±10,±15,±18,±30,±45,±90
Using Synthetic Division
We find that 1 and 3 are not zeros but that 2 is a zero and that P factors as
x4−x3−23x2−3x+90=(x−2)(x3+x2−21x−45)
We now factor the quotient x3+x2−21x−45. Its possible zeros are the divisors of −45, namely
±1,±3,±5,±9,±15,±45
Using Synthetic Division
We find that −3 is a zero and that P factors as
x4−x3−23x2−3x+90=(x−2)(x+3)(x2−2x−15)
We now factor x2−2x−15 using trial and error, so
x4−x3−23x2−3x+90=(x−2)(x+3)(x+3)(x−5)
The zeros of P are 2,5 and −3.
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