# 3.6 Zeros of polynomial functions  (Page 4/14)

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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 $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is a polynomial function of degree four, and $\text{\hspace{0.17em}}f\left(x\right)=0.\text{\hspace{0.17em}}$ The Fundamental Theorem of Algebra states that there is at least one complex solution, call it $\text{\hspace{0.17em}}{c}_{1}.\text{\hspace{0.17em}}$ By the Factor Theorem, we can write $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ as a product of $\text{\hspace{0.17em}}x-{c}_{\text{1}}\text{\hspace{0.17em}}$ and a polynomial quotient. Since $\text{\hspace{0.17em}}x-{c}_{\text{1}}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}{c}_{\text{2}}.\text{\hspace{0.17em}}$ So we can write the polynomial quotient as a product of $\text{\hspace{0.17em}}x-{c}_{\text{2}}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$

## 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 $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is a polynomial of degree $\text{\hspace{0.17em}}n>0,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a non-zero real number, then $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ has exactly $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ linear factors

$f\left(x\right)=a\left(x-{c}_{1}\right)\left(x-{c}_{2}\right)...\left(x-{c}_{n}\right)$

where $\text{\hspace{0.17em}}{c}_{1},{c}_{2},...,{c}_{n}\text{\hspace{0.17em}}$ are complex numbers. Therefore, $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ has $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}f\left(x\right)=3{x}^{3}+9{x}^{2}+x+3.\text{\hspace{0.17em}}$

The Rational Zero Theorem tells us that if $\text{\hspace{0.17em}}\frac{p}{q}\text{\hspace{0.17em}}$ is a zero of $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is a factor of 3 and $\text{\hspace{0.17em}}q\text{\hspace{0.17em}}$ is a factor of 3.

The factors of 3 are $±1\text{\hspace{0.17em}}$ and $±3.\text{\hspace{0.17em}}$ The possible values for $\text{\hspace{0.17em}}\frac{p}{q},\text{\hspace{0.17em}}$ and therefore the possible rational zeros for the function, are 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.

Dividing by $\text{\hspace{0.17em}}\left(x+3\right)\text{\hspace{0.17em}}$ gives a remainder of 0, so –3 is a zero of the function. The polynomial can be written as

$\left(x+3\right)\left(3{x}^{2}+1\right)$

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

The zeros of $f\left(x\right)$ are –3 and $\text{\hspace{0.17em}}±\frac{i\sqrt{3}}{3}.$

Find the zeros of $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{3}+5{x}^{2}-11x+4.$

The zeros are

## 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 $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ will have $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the polynomial function into $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\left(x-c\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is a complex number.

can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
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."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator