Wednesday, August 27, 2014

College Algebra, Chapter 4, 4.4, Section 4.4, Problem 48

Determine all the real zeros of the polynomial P(x)=x4+2x32x23x+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+2x32x23x+2=(x1)(x3+3x2+x2)

We now factor the quotient x3+3x2+x2 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+2x32x23x+2=(x1)(x+2)(x2+x1)

We now factor the quotient x2+x1 using the quadratic formula


x=b±b24ac2ax=1±(1)24(1)(1)2(1)x=1±52


The zeros of P are 1,2,1+52 and 152

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