E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals 3.4 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

KFUPM - Prep Year Math Program (c) 20013 All Right Reserved

3.4


Zeros of Polynomial Functions

Fundamental Theorem of Algebra Conjugate Zero Theorem End behavior of a Polynomial Intermediate Value Theorem Bisection Method



Fundamental Theorem of Algebra

A complex number is said to be a zero of a polynomial function

if and only if . That is is a solution (or sometimes called a root) of the polynomial equation

.A polynomial equation of degree may have real or complex roots and some of them may be repeated.Every polynomial function of degree has at least one complex zero.



Repeated Zero Theorem

A polynomial function

of degree , has exactly complex zeros, some of which may be repeated. Furthermore, if are the distinct zeros of then

where is the number of times is repeated as a zero of , and

.



Conjugate Zero Theorem

Each zero of the polynomial function

may be real or complex, and if is a non-real complex zero of

then is a non-real complex zero of provided that are all real numbers.



Example 1 Find the zeros and state the multiplicity and the degree of

Zeros Multiplicity

Therefore, the degree of is 9.

𝑃 (𝑥 )=𝑥 (𝑥−1 )2 (𝑥+2 𝑖 ) (𝑥−2 𝑖 ) (𝑥+5 ) (𝑥+2 )3



Example 2 Find and -intercepts, determine the far behavior, state whether the graph touches or crosses the-axis

𝑃 (𝑥 )=𝑥 (𝑥−1 )2 (𝑥+2 𝑖 ) (𝑥−2 𝑖 ) (𝑥+5 ) (𝑥+2 )3

The y-intercept is The x-intercepts are:

As x goes to , goes to (graph raises up). And as x goes to , goes to (graph falls down). Furthermore, the graph crosses the x-axis at , and touches the x-axis at .



Example 3 Sketch the graph of the polynomial function

𝑃 (𝑥 )=𝑥 (𝑥−1 )2 (𝑥+2 𝑖 ) (𝑥−2 𝑖 ) (𝑥+5 ) (𝑥+2 )3



Example 4 Find a polynomial with real coefficients which satisfies the conditions

, has zeros at of multiplicity , at of multiplicity , and at of multiplicity .

But , then



The polynomial has a cross at , so is a factor of . And the polynomial has a touch at so is a factor also of .

y-intercept is , so

Example 5 Find a polynomial of least degree that has the given graph



Intermediate Value Theorem

For any polynomial , if there are two numbers and such that

,then must have at least one real zero of odd multiplicity between and . Such a zero is an x-intercept of at which the graph of crosses the x-axis.

𝒛𝑎 𝑏𝑃 (𝑎 )

𝑃 (𝑏 )



Example 6 Show that

(a) has at least one real zero between 1 and 2(b) has at least two real zeros between 0 and 2.

(a) Since and ,it follows that has a simple zero between 1 and 2.

(b) Since and ,then has at least one real zero of odd multiplicity between 0 and 2. Since the number of complex zeros must be even, must have at least two real zeros.



Bisection method

Assume that we know that has exactly one zero of odd multiplicity within an interval , and that . Starting by the initial interval , the method approximates by constructing a sequence of smaller intervals each of which contains . We keep applying the method till the interval is small enough and we then take its midpoint as our approximation.



Example 7 Assume that the zero of within is unique.

Find an interval of length which contains and then find an approximation of , estimate the error in this approximation.

Then the zero is within the interval .



Example 7 continue

Now take and

then the zero is within the interval .Approximate by and the is less than , half the length of the interval.

Documents

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals 3.4 E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals