C H A P T E R 2 Polynomial and Rational Functions

Section 2.5 Zeros of Polynomial Functions
Section 2.5 Zeros of Polynomial Functions 187
■ You should know that if f is a polynomial of degree then f has at least one zero in the complex number system.
■ You should know the Linear Factorization Theorem.
■ You should know the Rational Zero Test.
■ You should know shortcuts for the Rational Zero Test. Possible rational zeros
(a) Use a graphing or programmable calculator.
(b) Sketch a graph.
(c) After finding a root, use synthetic division to reduce the degree of the polynomial.
■ You should know that if is a complex zero of a polynomial f, with real coefficients, then is also a complex
zero of f.
■ You should know the difference between a factor that is irreducible over the rationals (such as ) and a factor that is
irreducible over the reals (such as x ).
■ You should know Descartes’s Rule of Signs. (For a polynomial with real coefficients and a non-zero constant term.)
(a) The number of positive real zeros of f is either equal to the number of variations of sign of f or is less than
that number by an even integer.
(b) The number of negative real zeros of f is either equal to the number of variations in sign of fx or is less than
that number by an even integer.
(c) When there is only one variation in sign, there is exactly one positive (or negative) real zero.
■ You should be able to observe the last row obtained from synthetic division in order to determine upper or lower bounds.
(a) If the test value is positive and all of the entries in the last row are positive or zero, then the test value is an upper
bound.
(b) If the test value is negative and the entries in the last row alternate from positive to negative, then the test value is a
lower bound. (Zero entries count as positive or negative.)
2 x
9
2 n > 0,
factors of constant term
factors of leading coefficient
a bi
a bi
7
Vocabulary Check
1.
3.
1. Fundamental Theorem of Algebra 2. Linear Factorization Theorem 3. Rational Zero
4. conjugate 5. irreducible; reals 6. Descarte’s Rule of Signs
7. lower; upper
f x xx 6 2 2.
The zeros are: x 0, x 6
gx x 2x 4 3 4.
The zeros are: x 2, x 4
6. ht t 3t 2t 3it 3i
The four zeros are: 3, 2, ±3i
7.
f x x 2 x 3x 2 1 x 2 x 3x 1x 1
The five zeros are: 0, 0, 3, ±1
f x x 5x 82 5. f x x 6x ix i
The three zeros are: 5, 8, 8
fx x 3 3x 2 x 3
Possible rational zeros: ±1, ±3
The three zeros are:
x 6, x i, x i
Zeros shown on graph: 3, 1, 1