Friday, August 24, 2018

College Algebra, Chapter 4, 4.4, Section 4.4, Problem 28

Determine all rational zeros of the polynomial P(x)=x4x323x23x+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

x4x323x23x+90=(x2)(x3+x221x45)

We now factor the quotient x3+x221x45. 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

x4x323x23x+90=(x2)(x+3)(x22x15)

We now factor x22x15 using trial and error, so

x4x323x23x+90=(x2)(x+3)(x+3)(x5)

The zeros of P are 2,5 and 3.

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