Zeros of Polynomial Functions

MATH 109 - PrecalculusS. Rook

Overview

• Section 2.5 in the textbook:– Fundamental Theorem of Algebra– Rational Zeros Test– Complex Zeros

2

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

• Fundamental Theorem of Algebra: The polynomial function f(x) has EXACTLY n zeros where n is the degree of f(x)– The zeros may be:• Real:– e.g. f(x) = x2 – 1 = (x + 1)(x – 1) → x = -1, 1 (2 zeros)

• Imaginary– e.g. f(x) = x2 + 1 = (x + i)(x – i) → x = -i, i (2 zeros)

• Complex– e.g. f(x) = x2 – x + 1 → (2 zeros)

4

2

3

2

1,

2

3

2

1 iix

Fundamental Theorem of Algebra (Continued)

• A mixture of the two– e.g. f(x) = x4 – 1 = (x2 + 1)(x2 – 1) =

(x + i)(x – i)(x + 1)(x – 1) → x = -i, i, -1, 1 (4 zeros)– A zero x = a with multiplicity k contributes k zeros to

the polynomial• e.g. f(x) = x3 – x2 – x + 1 = (x – 1)2(x + 1) →

x = -1, 1 [multiplicity 2] (3 zeros)

5

Fundamental Theorem of Algebra (Example)

Ex 1: Given the following zeros, i) write each as a linear factor ii) combine the linear factors to obtain a polynomial f(x) with the given zeros

x = 0, 2 + i, 2 – i

6

Rational Zeros Test

8

Rational Zeros Test• Sometimes it is not always so simple to find the zeros of a

polynomial– Especially when the degree goes beyond 2

• Finding zeros is an important aspect of algebra• Rational Zeros Test: If a polynomial is in the form f(x) =

anxn + an-1xn-1 + … + a1x + a0, then p / q where p is a factor of a0 (constant term) and q is a factor of an (leading term) represents ALL of the possible RATIONAL zeros of f(x)– i.e. List all the factors of p & q and write down all possible

combinations of p / q being careful to exclude duplicates• e.g. p = 1, q = 2 ± {1} / ± {1, 2} ±{1, ½}

f(x) must be in DESCENDING degree

Rational Zeros Test (Continued)

• To find the zeros of a polynomial function f(x) with a degree higher than 2:– Factor any possible GCF from the polynomial– Apply the Rational Zeros Test to list the potential RATIONAL

zeros of f(x)– Use the potential zeros along with the Remainder Theorem to

find a zero x = c such that f(c) = 0• Can be time-consuming because we have to use trial and

error– Apply synthetic division using the zero x = c– Apply the appropriate strategy if the resulting polynomial has a

degree of two or less; otherwise, repeat the Rational Zeros Test

9

Rational Zeros Test (Example)

Ex 2: Find all the real zeros of the function:

a) f(x) = x3 + 2x2 – x – 2

b) g(x) = x4 – x3 – 2x – 4

c) h(x) = x3 – 6x2 + 11x – 6

10

More on the Rational Zeros Test

• Bear in mind that the Rational Zeros Test only helps to find the RATIONAL zeros– i.e. CANNOT be used to find zeros that are:

• Irrational• Imaginary

– Apply the Rational Zeros Test to find all of the RATIONAL zeros• Any irrational or imaginary solutions will come out in the end • If no rational zeros exist, then we need to find another way

to extract the zeros from the polynomial

11

More on the Rational Zeros Test (Example)

Ex 3: Find all the real zeros of the function

a) f(x) = x3 – 2x2 – x

b) g(x) = x3 + x2 – 3x – 3

12

Complex Zeros

Complex Zeros

• Conjugate Zeros Theorem: Let f(x) be a polynomial function with REAL coefficients. If x = a + bi is a zero of f(x), then so is its conjugate x = a – bi and vice versa– Recall that the product of a complex number and

its conjugate produces a real number– Likewise, the product of a complex factor and its

conjugate produces a factor with real coefficients• e.g. (x – i)(x + i) = x2 – i2 = x2 + 1

14

Complex Zeros (Continued)

• Given a complex zero of a polynomial function f(x):– Use synthetic division on f(x) with the given complex

zero• This will more than likely produce a function with complex

coefficients

– Use synthetic division on the result with the conjugate of the complex zero• This will produce a polynomial with real coefficients

– Use either the Rational Zeros Test or other strategies to break apart the polynomial

15

Complex Zeros (Example)

Ex 4: Use the given zero to find all the zeros of the function:

a) f(x) = 4x3 + 23x2 + 34x – 10; x = -3 + i

b) g(x) = 2x3 + 3x2 + 50x + 75; x = 5i

16

Complex Zeros (Example)

Ex 5: Find all the zeros of the function AND write the polynomial as a product of linear factors:

a) f(x) = x4 – 4x3 + 8x2 – 16x + 16

b) g(x) = x3 – x + 6

17

Summary

• After studying these slides, you should be able to:– Understand the Fundamental Theorem of Algebra in terms

of the number of zeros of a polynomial function– Write a list of zeros as linear factors and to combine a list of

linear factors into a polynomial– Apply the Rational Zeros test to find the zeros of a

polynomial• Additional Practice

– See the list of suggested problems for 2.5• Next lesson

– Rational Functions (Section 2.6)

18