Determine all the real zeros of the polynomial P(x)=x4+2x3−2x2−3x+2. Use the quadratic formula if necessary.
The leading coefficient of P is 1, so all the rational zeros are integers. They are the divisors of constant term 2. Thus, the possible candidates are
±1,±2
Using Synthetic Division,
We find that 1 is a zero and that P factors as
x4+2x3−2x2−3x+2=(x−1)(x3+3x2+x−2)
We now factor the quotient x3+3x2+x−2 and its possible zeros are
±1,±2
Using Synthetic Division,
We find that −1,1 and 2 are not zeros but that −2 is a zero and that P factors as
x4+2x3−2x2−3x+2=(x−1)(x+2)(x2+x−1)
We now factor the quotient x2+x−1 using the quadratic formula
x=−b±√b2−4ac2ax=−1±√(1)2−4(1)(−1)2(1)x=−1±√52
The zeros of P are 1,−2,−1+√52 and −1−√52
No comments:
Post a Comment