



Key equations
general form of a polynomial function 
$$f(x)={a}_{n}{x}^{n}+\mathrm{...}+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$$ 
Key concepts
 A power function is a variable base raised to a number power. See
[link] .
 The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
 The end behavior depends on whether the power is even or odd. See
[link] and
[link] .
 A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See
[link] .
 The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See
[link] .
 The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See
[link] and
[link] .
 A polynomial of degree
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ will have at most
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$
x intercepts and at most
$\text{\hspace{0.17em}}n1\text{\hspace{0.17em}}$ turning points. See
[link] ,
[link] ,
[link] ,
[link] , and
[link] .
Section exercises
Verbal
Explain the difference between the coefficient of a power function and its degree.
The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.
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In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.
As
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases without bound, so does
$\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ As
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases without bound, so does
$\text{\hspace{0.17em}}f\left(x\right).$
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What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As
$\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty \text{\hspace{0.17em}}$ and as
$\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty .\text{\hspace{0.17em}}$
The polynomial function is of even degree and leading coefficient is negative.
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Algebraic
For the following exercises, identify the function as a power function, a polynomial function, or neither.
For the following exercises, find the degree and leading coefficient for the given polynomial.
For the following exercises, determine the end behavior of the functions.
$f\left(x\right)={x}^{4}$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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$f\left(x\right)={x}^{4}$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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$f(x)=2{x}^{4}3{x}^{2}+x1$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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$f(x)={x}^{2}\left(2{x}^{3}x+1\right)$
$\text{As}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(x)\to \infty ,\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f(x)\to \infty $
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For the following exercises, find the intercepts of the functions.
$f\left(t\right)=2\left(t1\right)\left(t+2\right)(t3)$
y intercept is
$\text{\hspace{0.17em}}(0,12),\text{\hspace{0.17em}}$
t intercepts are
$\text{\hspace{0.17em}}(1,0);(\u20132,0);\text{and}(3,0).$
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y intercept is
$\text{\hspace{0.17em}}(0,16).\text{\hspace{0.17em}}$
x intercepts are
$\text{\hspace{0.17em}}(2,0)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}(2,0).$
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$f(x)=x\left({x}^{2}2x8\right)$
y intercept is
$\text{\hspace{0.17em}}(0,0).\text{\hspace{0.17em}}$
x intercepts are
$\text{\hspace{0.17em}}(0,0),(4,0),\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left(2,0\right).$
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Graphical
For the following exercises, determine the least possible degree of the polynomial function shown.
Questions & Answers
bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
remove any signs and collect terms of 2(8a3bc)
is that a real answer
Joeval
x²2x+94x²+12x20
3x²+10x+11
Miranda
x²2x+94x²+12x20
3x²+10x+11
Miranda
(X22X+8)4(X23X+5)=0 ?
master
The anwser is imaginary
number if you want to know The anwser of the expression
you must arrange The expression and use quadratic formula To find the
answer
master
The anwser is imaginary
number if you want to know The anwser of the expression
you must arrange The expression and use quadratic formula To find the
answer
master
X22X+84X2+12X20=0
(X24X2)+(2X+12X)+(20+8)= 0
3X2+10X12=0
3X210X+12=0
Use quadratic formula To find the answer
answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x22x+84x2+12x20
x24x22x+12x+820
3x2+10x12
now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and
x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
given 4cot thither +3=0and 0°<thither <180°
use a sketch to determine the value of the following
a)cos thither
nothing up todat yet
Miranda
aap konsi country se ho
jai
which language is that
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math.
I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
what is the solution of the given equation?
please where is the equation
Miranda
Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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