# 3.4 Graphs of polynomial functions  (Page 2/13)

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## Finding the x -intercepts of a polynomial function by factoring

Find the x -intercepts of $\text{\hspace{0.17em}}f\left(x\right)={x}^{6}-3{x}^{4}+2{x}^{2}.$

We can attempt to factor this polynomial to find solutions for $\text{\hspace{0.17em}}f\left(x\right)=0.$

This gives us five $\text{\hspace{0.17em}}x\text{-}$ intercepts: $\text{\hspace{0.17em}}\left(0,0\right),\left(1,0\right),\left(-1,0\right),\left(\sqrt{2},0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-\sqrt{2},0\right).\text{\hspace{0.17em}}$ See [link] . We can see that this is an even function.

## Finding the x -intercepts of a polynomial function by factoring

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

Find solutions for $\text{\hspace{0.17em}}f\left(x\right)=0\text{\hspace{0.17em}}$ by factoring.

$\begin{array}{lllll}x+1=0\hfill & \text{or}\hfill & x-1=0\hfill & \text{or}\hfill & x-5=0\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=-1\hfill & \hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=1\hfill & \hfill & \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=5\hfill \end{array}$

There are three $\text{\hspace{0.17em}}x\text{-}$ intercepts: $\text{\hspace{0.17em}}\left(-1,0\right),\left(1,0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,0\right).\text{\hspace{0.17em}}$ See [link] .

## Finding the y - and x -intercepts of a polynomial in factored form

Find the $\text{\hspace{0.17em}}y\text{-}$ and x -intercepts of $\text{\hspace{0.17em}}g\left(x\right)={\left(x-2\right)}^{2}\left(2x+3\right).$

The y -intercept can be found by evaluating $\text{\hspace{0.17em}}g\left(0\right).$

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}g\left(0\right)={\left(0-2\right)}^{2}\left(2\left(0\right)+3\right)\\ =12\end{array}$

So the y -intercept is $\text{\hspace{0.17em}}\left(0,12\right).$

The x -intercepts can be found by solving $\text{\hspace{0.17em}}g\left(x\right)=0.$

${\left(x-2\right)}^{2}\left(2x+3\right)=0$

So the $\text{\hspace{0.17em}}x\text{-}$ intercepts are $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-\frac{3}{2},0\right).$

## Finding the x -intercepts of a polynomial function using a graph

Find the $\text{\hspace{0.17em}}x\text{-}$ intercepts of $\text{\hspace{0.17em}}h\left(x\right)={x}^{3}+4{x}^{2}+x-6.$

This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.

Looking at the graph of this function, as shown in [link] , it appears that there are x -intercepts at $\text{\hspace{0.17em}}x=-3,-2,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.$

We can check whether these are correct by substituting these values for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and verifying that

$h\left(-3\right)=h\left(-2\right)=h\left(1\right)=0.$

Since $\text{\hspace{0.17em}}h\left(x\right)={x}^{3}+4{x}^{2}+x-6,\text{\hspace{0.17em}}$ we have:

Each $\text{\hspace{0.17em}}x\text{-}$ intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.

Find the $\text{\hspace{0.17em}}y\text{-}$ and x -intercepts of the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{4}-19{x}^{2}+30x.$

y -intercept $\text{\hspace{0.17em}}\left(0,0\right);\text{\hspace{0.17em}}$ x -intercepts $\text{\hspace{0.17em}}\left(0,0\right),\left(–5,0\right),\left(2,0\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,0\right)$

## Identifying zeros and their multiplicities

Graphs behave differently at various $\text{\hspace{0.17em}}x\text{-}$ intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.

Suppose, for example, we graph the function

$f\left(x\right)=\left(x+3\right){\left(x-2\right)}^{2}{\left(x+1\right)}^{3}.$

Notice in [link] that the behavior of the function at each of the $\text{\hspace{0.17em}}x\text{-}$ intercepts is different.

The $\text{\hspace{0.17em}}x\text{-}$ intercept $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ is the solution of equation $\text{\hspace{0.17em}}\left(x+3\right)=0.\text{\hspace{0.17em}}$ The graph passes directly through the $\text{\hspace{0.17em}}x\text{-}$ intercept at $\text{\hspace{0.17em}}x=-3.\text{\hspace{0.17em}}$ The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich