# 5.2 Power functions and polynomial functions  (Page 4/19)

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## Identifying the degree and leading coefficient of a polynomial function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

$\begin{array}{ccc}\hfill f\left(x\right)& =& 3+2{x}^{2}-4{x}^{3}\hfill \\ \hfill g\left(t\right)& =& 5{t}^{2}-2{t}^{3}+7t\hfill \\ h\left(p\right)\hfill & =& 6p-{p}^{3}-2\hfill \end{array}$

For the function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ the highest power of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is 3, so the degree is 3. The leading term is the term containing that degree, $\text{\hspace{0.17em}}-4{x}^{3}.\text{\hspace{0.17em}}$ The leading coefficient is the coefficient of that term, $\text{\hspace{0.17em}}-4.$

For the function $\text{\hspace{0.17em}}g\left(t\right),\text{\hspace{0.17em}}$ the highest power of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}5,\text{\hspace{0.17em}}$ so the degree is $\text{\hspace{0.17em}}5.\text{\hspace{0.17em}}$ The leading term is the term containing that degree, $\text{\hspace{0.17em}}5{t}^{5}.\text{\hspace{0.17em}}$ The leading coefficient is the coefficient of that term, $\text{\hspace{0.17em}}5.$

For the function $\text{\hspace{0.17em}}h\left(p\right),\text{\hspace{0.17em}}$ the highest power of $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}$ so the degree is $\text{\hspace{0.17em}}3.\text{\hspace{0.17em}}$ The leading term is the term containing that degree, $\text{\hspace{0.17em}}-{p}^{3}.\text{\hspace{0.17em}}$ The leading coefficient is the coefficient of that term, $\text{\hspace{0.17em}}-1.$

Identify the degree, leading term, and leading coefficient of the polynomial $\text{\hspace{0.17em}}f\left(x\right)=4{x}^{2}-{x}^{6}+2x-6.$

The degree is 6. The leading term is $\text{\hspace{0.17em}}-{x}^{6}.\text{\hspace{0.17em}}$ The leading coefficient is $\text{\hspace{0.17em}}-1.$

## Identifying end behavior of polynomial functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See [link] .

Polynomial Function Leading Term Graph of Polynomial Function
$f\left(x\right)=5{x}^{4}+2{x}^{3}-x-4$ $5{x}^{4}$
$f\left(x\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}$ $-2{x}^{6}$
$f\left(x\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1$ $3{x}^{5}$
$f\left(x\right)=-6{x}^{3}+7{x}^{2}+3x+1$ $-6{x}^{3}$

## Identifying end behavior and degree of a polynomial function

Describe the end behavior and determine a possible degree of the polynomial function in [link] .

As the input values $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ get very large, the output values $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ increase without bound. As the input values $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ get very small, the output values $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ decrease without bound. We can describe the end behavior symbolically by writing

In words, we could say that as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ values approach infinity, the function values approach infinity, and as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

Describe the end behavior, and determine a possible degree of the polynomial function in [link] .

As It has the shape of an even degree power function with a negative coefficient.

## Identifying end behavior and degree of a polynomial function

Given the function $\text{\hspace{0.17em}}f\left(x\right)=-3{x}^{2}\left(x-1\right)\left(x+4\right),\text{\hspace{0.17em}}$ express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Obtain the general form by expanding the given expression for $\text{\hspace{0.17em}}f\left(x\right).$

$\begin{array}{ccc}\hfill f\left(x\right)& =& -3{x}^{2}\left(x-1\right)\left(x+4\right)\hfill \\ & =& -3{x}^{2}\left({x}^{2}+3x-4\right)\hfill \\ & =& -3{x}^{4}-9{x}^{3}+12{x}^{2}\hfill \end{array}$

The general form is $\text{\hspace{0.17em}}f\left(x\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}.\text{\hspace{0.17em}}$ The leading term is $\text{\hspace{0.17em}}-3{x}^{4};\text{\hspace{0.17em}}$ therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is

Cos45/sec30+cosec30=
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
what is one-to-one function
what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1