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Using the fundamental theorem of algebra

Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.

Suppose f is a polynomial function of degree four, and f ( x ) = 0. The Fundamental Theorem of Algebra states that there is at least one complex solution, call it c 1 . By the Factor Theorem, we can write f ( x ) as a product of x c 1 and a polynomial quotient. Since x c 1 is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it c 2 . So we can write the polynomial quotient as a product of x c 2 and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of f ( x ) .

The Fundamental Theorem of Algebra    States that, if f(x) Is a polynomial of degree n>0 , then f(x) Has at least one complex zero.

We can use this theorem to argue that, if f ( x ) is a polynomial of degree n > 0 , and a is a non-zero real number, then f ( x ) has exactly n linear factors

f ( x ) = a ( x c 1 ) ( x c 2 ) ... ( x c n )

where c 1 , c 2 , ... , c n are complex numbers. Therefore, f ( x ) has n roots if we allow for multiplicities.

Does every polynomial have at least one imaginary zero?

No. A complex number is not necessarily imaginary. Real numbers are also complex numbers.

Finding the zeros of a polynomial function with complex zeros

Find the zeros of f ( x ) = 3 x 3 + 9 x 2 + x + 3.

The Rational Zero Theorem tells us that if p q is a zero of f ( x ) , then p is a factor of 3 and q is a factor of 3.

p q = factor of constant term factor of leading coefficient     = factor of 3 factor of 3

The factors of 3 are ±1 and ±3. The possible values for p q , and therefore the possible rational zeros for the function, are ±3 , ±1, and  ± 1 3 . We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Let’s begin with –3.

3 3 9 1 3 9 0 3      3     0   1    0

Dividing by ( x + 3 ) gives a remainder of 0, so –3 is a zero of the function. The polynomial can be written as

( x + 3 ) ( 3 x 2 + 1 )

We can then set the quadratic equal to 0 and solve to find the other zeros of the function.

3 x 2 + 1 = 0         x 2 = 1 3          x = ± 1 3 = ± i 3 3

The zeros of f ( x ) are –3 and ± i 3 3 .

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Find the zeros of f ( x ) = 2 x 3 + 5 x 2 11 x + 4.

The zeros are –4,  1 2 ,  and 1 .

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Using the linear factorization theorem to find polynomials with given zeros

A vital implication of the Fundamental Theorem of Algebra    , as we stated above, is that a polynomial function of degree n will have n zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into n factors. The Linear Factorization Theorem    tells us that a polynomial function will have the same number of factors as its degree, and that each factor will be in the form ( x c ) , where c is a complex number.

Questions & Answers

what is f(x)=
Karim Reply
I don't understand
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
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It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Now it shows, go figure?
what is this?
unknown Reply
i do not understand anything
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I've been struggling so much through all of this. my final is in four weeks 😭
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is there any question in particular?
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Sure, are you in high school or college?
Hi, apologies for the delayed response. I'm in college.
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
I done know
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I had just woken up when i got this message
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
i have a question.
how do you find the real and complex roots of a polynomial?
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
@Nare please let me know if you can solve it.
I have a question
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The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
if not then how would I find it from a graph
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
you could also do it with two consecutive minimum points or x-intercepts
I will try that thank u
Case of Equilateral Hyperbola
Jhon Reply
f(x)=4x+2, find f(3)
f(3)=4(3)+2 f(3)=14
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
more than 6000
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
Practice Key Terms 6

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