e - book for college algebra king fahd university of petroleum & minerals 3.4 e - book for...
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E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals Repeated Zero Theorem KFUPM - Prep Year Math Program (c) 2009 All Right ReservedTRANSCRIPT
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 20013 All Right Reserved
3.4
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
Zeros of Polynomial Functions
Fundamental Theorem of Algebra Conjugate Zero Theorem End behavior of a Polynomial Intermediate Value Theorem Bisection Method
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 .
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 )
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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 .
E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals
KFUPM - Prep Year Math Program (c) 2009 All Right Reserved
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.