The polynomial P(x)=−x4+10x2+8x−8.
a.) Find all the real zeros of P
The leading coefficient of P is −1, so all rational zeros are integers. They are divisors of constant term −8. Thus, the possible zeros are
±1,±2,±4,±8
Using Synthetic Division
We find that 1,2,4 and −1 are not zeros but that −2 is zero and that P factors as
−x4+10x2+8x−8=(x+2)(−x3+2x2+6x−4)
We now factor the quotient −x3+2x2+6x−4. Its possible zeros are
±1,±2,±4
Using Synthetic Division
We find that −2 is a zero and that P factors as
−x4+10x2+8x−8=(x+2)(x+2)(−x2+4x−2)
We now factor the quotient −x2+4x−2 using Quadratic Formula, we get
x=−b±√b2−4ac2ax=−4±√(4)2−4(−1)(−2)2(−1)x=2±√2
The zeros of P are −2,2+√2 and 2−√2.
b.) Sketch the graph of P
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