Zeros of Polynomial Functions

Zeros of Polynomial Functions

Advanced MathSection 3.4

Advanced Math 3.4 - 3.5 2

Number of zeros

• Any nth degree polynomial can have at most n real zeros

• Using complex numbers, every nth degree polynomial has precisely n zeros (real or imaginary)


Fundamental Theorem of Algebra

• If f(x) is a polynomial of degree n, where n > 0,

• then f has at least one zero in the complex number system


Linear Factorization Theorem

• If f(x) is a polynomial of degree n, where n > 0,

• then f has precisely n linear factors

1 2n nf x a x c x c x c

1 2where , , are complex numbersnc c c


Linear Factorization Theorem applied

1st degree: 5 has exactly one zero -5f x x x

22nd degree: 10 25 has exactly two zeros

5 5 5 and 5

(multiplicity counts: 5 is a repeated zero)

f x x x

f x x x x x

3

2

3rd degree: +9x has exactly three zeros

9 3 3 0, 3

f x x

f x x x x x i x i x x i


Example

• Find all zeros4 1x


Rational Zero Test

1 21 2 1 0If the polynomial

has coefficients, every rational zero of has the form

rational zero

where and have no common factors other than 1, and a factor of the

n nn nf x a x a x a x a x a

integer fpq

p qp

0 constant term a factor of the leading coefficient n

aq a


Using the rational zero test

• List all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficientfactors of constant termpossible rational zeros =

factors of leading coefficent

• Use trial-and-error to determine which, if any are actual zeros of the polynomial

• Can use table on graphing calculator to speed up calculations


Example

• Use the Rational Zero Test to find the rational zeros

3 24 4 16f x x x x


Using synthetic division

• Test all factors to see if the remainder is zero

• Can also use graphing calculator to estimate zeros, then only check possibilities near your estimate

3 28 40 525f x x x x


Examples

• Find all rational zeros

3 22 3 8 3f x x x x

3 22 3 1f x x x


Conjugate pairs

• If the polynomial has real coefficients,

• then zeros occur in conjugate pairs• If a + bi is a zero, then a – bi also

is a zero.

a bia bi


Example:

• Find a fourth-degree polynomial function with real coefficients that has zeros -2, -2, and 4i


Factors of a Polynomial

• Even if you don’t want to use complex numbers

• Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros


Quadratic factors

• If they can’t be factored farther without using complex numbers, they are irreducible over the reals

2 1x x i x i


Quadratic factors

• If they can’t be factored farther without using irrational numbers, they are irreducible over the rationals– These are reducible over the reals

2 2 2 2x x x


Finding zeros of a polynomial function

• If given a complex factor– Its conjugate must be a factor– Multiply the two conjugates – this will give

you a real zero– Use long division or synthetic division to

find more factors• If not given any factors

– Use the rational zero test to find rational zeros

– Factor or use the quadratic formula to find the rest


Examples

• Use the given zero to find all zeros of the function 3 2 9 9, zero 3f x x x x i

3 24 23 34 10, zero 3f x x x x i


Examples

• Find all the zeros of the function and write the polynomial as a product of linear factors

2 4 1h x x x

3 26 13 10g x x x x

4 210 9f x x x


Descartes’s Rule of Signs

– A variation in sign means that two consecutive coefficients have opposite signs

• For a polynomial with real coefficients and a constant term,

• The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer

• The number of negative real zeros is either equal to the variations in sign of f(-x) or less than that number by an even integer.


Examples

• Determine the possible numbers of positive and negative zeros

3 22 3 1g x x x

3 23 2 3f x x x x


Upper Bound Rule

– When using synthetic division• If what you try isn’t a factor, but• The number on the outside of the

synthetic division is positive – And each number in the answer is

either positive or zero– then the number on the outside is an

upper bound for the real zeros


Lower Bound Rule

– When using synthetic division• If what you try isn’t a factor, but• The number on the outside of the

synthetic division is negative – The numbers in the answer are

alternately positive and negative (zeros can count as either)

– then the number on the outside is a lower bound for the real zeros


Examples

• Use synthetic division to verify the upper and lower bounds of the real zeros

4 34 15

: 4: 1

f x x x

Upper xLower x

3 22 3 12 8

: 4: 3

f x x x x

Upper xLower x

Mathematical Modeling and Variation

Advanced MathSection 3.5


Two basic types of linear models

• y-intercept is nonzero

• y-intercept is zero

y mx b

y mx


Direct Variation

• Linear• k is slope• y varies directly as x• y is directly proportional to x

for some nonzero constant ky kx


Direct Variation as an nth power

• y varies directly as the nth power of x

• y is directly proportional to the nth power of x

for some constant kny kx


Inverse Variation

• Hyperbola (when k is nonzero)• y varies inversely as x• y is inversely proportional to x

for some constant kkyx


Inverse Variation as an nth power

• y varies inversely as the nth power of x

• y is inversely proportional to the nth power of x

for some constant kn

kyx


Joint Variation

• Describes two different direct variations

• z varies jointly as x and y• z is jointly proportional to x and y

for some constant kz kxy


Joint Variation as an nth and mth power

• z varies jointly as the nth power of x and the mth power of y

• z is jointly proportional to the nth power of x and the mth power of y

y for some constant kn mz kx


Examples

• Find a math model representing the following statements and find the constants of proportionality

• A varies directly as r2.– When r = 3, A = 9p

• y varies inversely as x– When x = 25, y = 3

• z varies jointly as x and y– When x = 4 and y = 8, z = 64

Zeros of Polynomial Functions

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