# 5.8 Modeling using variation  (Page 7/14)

 Page 7 / 14

## Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.

$\frac{{x}^{3}-2{x}^{2}+4x+4}{x-2}$

$\text{\hspace{0.17em}}{x}^{2}+4\text{\hspace{0.17em}}$ with remainder 12

$\frac{3{x}^{4}-4{x}^{2}+4x+8}{x+1}$

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

$\frac{{x}^{3}-2{x}^{2}+5x-1}{x+3}$

${x}^{2}-5x+20-\frac{61}{x+3}$

$\frac{{x}^{3}+4x+10}{x-3}$

$\frac{2{x}^{3}+6{x}^{2}-11x-12}{x+4}$

$2{x}^{2}-2x-3$ , so factored form is $\left(x+4\right)\left(2{x}^{2}-2x-3\right)$

$\frac{3{x}^{4}+3{x}^{3}+2x+2}{x+1}$

## Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

$2{x}^{3}-3{x}^{2}-18x-8=0$

$3{x}^{3}+11{x}^{2}+8x-4=0$

$2{x}^{4}-17{x}^{3}+46{x}^{2}-43x+12=0$

$4{x}^{4}+8{x}^{3}+19{x}^{2}+32x+12=0$

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

${x}^{3}-3{x}^{2}-2x+4=0$

0 or 2 positive, 1 negative

$2{x}^{4}-{x}^{3}+4{x}^{2}-5x+1=0$

## Rational Functions

For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.

$f\left(x\right)=\frac{x+2}{x-5}$

Intercepts $\left(–2,0\right)\text{and}\left(0,-\frac{2}{5}\right)$ , Asymptotes $\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=1.$

$f\left(x\right)=\frac{{x}^{2}+1}{{x}^{2}-4}$

$f\left(x\right)=\frac{3{x}^{2}-27}{{x}^{2}+x-2}$

Intercepts (3, 0), (-3, 0), and $\text{\hspace{0.17em}}\left(0,\frac{27}{2}\right)\text{\hspace{0.17em}}$ , Asymptotes

$f\left(x\right)=\frac{x+2}{{x}^{2}-9}$

For the following exercises, find the slant asymptote.

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

$f\left(x\right)=\frac{2{x}^{3}-{x}^{2}+4}{{x}^{2}+1}$

For the following exercises, find the inverse of the function with the domain given.

$f\left(x\right)={\left(x-2\right)}^{2},\text{\hspace{0.17em}}x\ge 2$

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

$f\left(x\right)={\left(x+4\right)}^{2}-3,\text{\hspace{0.17em}}x\ge -4$

$f\left(x\right)={x}^{2}+6x-2,\text{\hspace{0.17em}}x\ge -3$

${f}^{-1}\left(x\right)=\sqrt{x+11}-3$

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

$f\left(x\right)=\sqrt{4x+5}-3$

${f}^{-1}\left(x\right)=\frac{{\left(x+3\right)}^{2}-5}{4},\text{\hspace{0.17em}}x\ge -3$

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

## Modeling Using Variation

For the following exercises, find the unknown value.

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the square of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If when find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=4.$

$y=64$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the square root of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ If when find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=4.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as the cube of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}z.\text{\hspace{0.17em}}$ If when $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=2,\text{\hspace{0.17em}}$ $y=6,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=3.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the square of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and inversely as the cube of $\text{\hspace{0.17em}}w.\text{\hspace{0.17em}}$ If when $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ $z=4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=2,\text{\hspace{0.17em}}$ $y=48,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=4,\text{\hspace{0.17em}}$ $z=5,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=3.$

For the following exercises, solve the application problem.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

148.5 pounds

The volume $\text{\hspace{0.17em}}V\text{\hspace{0.17em}}$ of an ideal gas varies directly with the temperature $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

## Chapter test

Give the degree and leading coefficient of the following polynomial function.

$f\left(x\right)={x}^{3}\left(3-6{x}^{2}-2{x}^{2}\right)$

Determine the end behavior of the polynomial function.

$f\left(x\right)=8{x}^{3}-3{x}^{2}+2x-4$

$As\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}as\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

$f\left(x\right)=-2{x}^{2}\left(4-3x-5{x}^{2}\right)$

Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

$f\left(x\right)={x}^{2}+2x-8$

$f\left(x\right)={\left(x+1\right)}^{2}-9$ , vertex $\text{\hspace{0.17em}}\left(-1,-9\right)$ , intercepts $\text{\hspace{0.17em}}\left(2,0\right);\left(-4,0\right);\text{\hspace{0.17em}}\left(0,-8\right)$

Given information about the graph of a quadratic function, find its equation.

Vertex $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and point on graph $\text{\hspace{0.17em}}\left(4,12\right).$

Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

60,000 square feet

Find all zeros of the following polynomial functions, noting multiplicities.

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

$f\left(x\right)=2{x}^{6}-12{x}^{5}+18{x}^{4}$

0 with multiplicity 4, 3 with multiplicity 2

Based on the graph, determine the zeros of the function and multiplicities.

Use long division to find the quotient.

$\frac{2{x}^{3}+3x-4}{x+2}$

$2{x}^{2}-4x+11-\frac{26}{x+2}$

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

$\frac{{x}^{4}+3{x}^{2}-4}{x-2}$

$\frac{2{x}^{3}+5{x}^{2}-7x-12}{x+3}$

$2{x}^{2}-x-4$ . So factored form is $\text{\hspace{0.17em}}\left(x+3\right)\left(2{x}^{2}-x-4\right)$

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f\left(x\right)=2{x}^{3}+5{x}^{2}-6x-9$

$f\left(x\right)=4{x}^{4}+8{x}^{3}+21{x}^{2}+17x+4$

$-\frac{1}{2}\text{\hspace{0.17em}}$ (has multiplicity 2), $\text{\hspace{0.17em}}\frac{-1±i\sqrt{15}}{2}\text{\hspace{0.17em}}$

$f\left(x\right)=4{x}^{4}+16{x}^{3}+13{x}^{2}-15x-18$

$f\left(x\right)={x}^{5}+6{x}^{4}+13{x}^{3}+14{x}^{2}+12x+8$

$\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}$ (has multiplicity 3), $\text{\hspace{0.17em}}±i$

Given the following information about a polynomial function, find the function.

It has a double zero at $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and zeros at $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-2\text{\hspace{0.17em}}$ . Its y -intercept is $\text{\hspace{0.17em}}\left(0,12\right).\text{\hspace{0.17em}}$

It has a zero of multiplicity 3 at $\text{\hspace{0.17em}}x=\frac{1}{2}\text{\hspace{0.17em}}$ and another zero at $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ . It contains the point $\text{\hspace{0.17em}}\left(1,8\right).$

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

Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

$8{x}^{3}-21{x}^{2}+6=0$

For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

$f\left(x\right)=\frac{x+4}{{x}^{2}-2x-3}$

Intercepts $\text{\hspace{0.17em}}\left(-4,0\right),\text{\hspace{0.17em}}\left(0,-\frac{4}{3}\right)\text{\hspace{0.17em}}$ , Asymptotes .

$f\left(x\right)=\frac{{x}^{2}+2x-3}{{x}^{2}-4}$

Find the slant asymptote of the rational function.

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

$y=x+4$

Find the inverse of the function.

$f\left(x\right)=\sqrt{x-2}+4$

$f\left(x\right)=3{x}^{3}-4$

${f}^{-1}\left(x\right)=\sqrt[3]{\frac{x+4}{3}}$

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

Find the unknown value.

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ $y=2.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=1.$

$y=18$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the cube root of $\text{\hspace{0.17em}}z.\text{\hspace{0.17em}}$ If when $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=27,\text{\hspace{0.17em}}$ $y=12,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=8.$

Solve the following application problem.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?

4 seconds

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
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
hello
Propessor
hi
Miranda
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
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
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
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Miranda
Jeffrey
Jeffrey
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
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function