College Algebra 9th txtbk

SECTION 4–3 Complex Zeros and Rational Zeros of Polynomials 295
So the exact zeros of P(x) are and 1 12.*
3
2
MATCHED PROBLEM 5
Find all zeros exactly for P(x) 3x 3 10x 2 5x 4.
EXAMPLE 6
Finding Rational and Imaginary Zeros
Find all zeros exactly for P(x) x 4 6x 3 14x 2 14x 5.
5
SOLUTION The possible rational zeros are 1 and 5. Examining the graph of P(x) (Fig. 7), we see
that 1 is a zero. Because the graph of P(x) does not appear to change sign at 1, this may
be a multiple root. Using synthetic division (details omitted), we find that
1
5
P(x) (x 1)(x 3 5x 2 9x 5)
The possible rational zeros of the reduced polynomial
Z Figure 7
1
5
Q(x) x 3 5x 2 9x 5
are 1 and 5. Examining the graph of Q(x) (Fig. 8), we see that 1 is a rational zero. After
a division, we have a quadratic reduced polynomial:
Q(x) (x 1)Q 1 (x) (x 1)(x 2 4x 5)
1
5
We use the quadratic formula to find the zeros of Q 1 (x):
Z Figure 8
5
x 2 4x 5 0
4 116 4(1)(5)
x
2
4 14 2 i
2
So the exact zeros of P(x) are 1 (multiplicity 2), 2 i, and 2 i.
MATCHED PROBLEM 6
Find all zeros exactly for P(x) x 4 4x 3 10x 2 12x 5.
REMARK
5
50
5
We were successful in finding all the zeros of the polynomials in Examples 5 and 6 because
we could find sufficient rational zeros to reduce the original polynomial to a quadratic. This
is not always possible. For example, the polynomial
P(x) x 3 6x 2
50
Z Figure 9 P(x) x 3 6x 2.
has no rational zeros, but does have an irrational zero at x 0.32748 (Fig. 9). The other
two zeros are imaginary. The techniques we have developed will not find the exact value of
these roots.
*By analogy with Theorem 3 (imaginary zeros of polynomials with real coefficients occur in conjugate pairs), it
can be shown that if r s1t is a zero of a polynomial with rational coefficients, where r, s, and t are rational
but t is not the square of a rational, then r s1t is also a zero.