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Resources

Conjugate root theorems

This is similar to Try It \#

8

in the OpenStax text.

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Given the function

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

, use the characteristics of polynomials and rational functions to describe its behavior and sketch the function.

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Enter the exact answers.

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Enter the intercepts as points,

(a, b)

x

-intercepts in increasing order of the

x

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x

-intercepts are

\square

\square

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y

\square

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The fields below accept a list of numbers or formulas separated by semicolons (e.g.

2 ; 4 ; 6

(a, b)

0

. The order of the list does not matter.

Find all solutions of the equation below.

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4^{3 \mathrm{x}+1}=7^{x}

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The solution(s) is/are

x=

\square

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(Simplify your answer. Use a comma to separate answers as needed. Round to four decimal places as needed.)

Find the volume of the solid generated by revolving the region bounded by the graphs of

y=x^{2}+3

y=x+9

x

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The volume of the solid is

\square

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(Type an exact answer, using

\pi

Find the volume of the solid generated by revolving the regions bounded by the lines and curves

\mathrm{y}=e^{(-1 / 8) \mathrm{x}}, \mathrm{y}=0

x=0

x=8

x

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The volume of the resulting solid is

\square

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(Type an exact answer, using

\pi

as needed. Use integers or fractions for any numbers in the expression.)

8

. A random sample of

200

female high school students in india showed that

110

are on Facebook A separate random sample of

150

male high school students in india showed that

69

are on Facebook. Let

p 1

p 2

be the proportion of all female and male high school students in India, respectively, who are on Facebook. Which of the following represents a

95 \%

confidence interval for

p 1

22

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\cos =2 \sqrt{\frac{\sin 6}{3}+\frac{\cos x}{\cos }}

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0.05=1.65 \sqrt{\frac{\operatorname{lan}}{2}}

Determine the qualities of the given set. (Select all that apply.)

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\left\{(x, y) \mid 9<x^{2}+y^{2}<25\right\}

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\square

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\square

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\square

simply-connected

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\square

none of the above

ilar triangles.

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2

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a. Identify any proportional relationships from the diagram.

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b. State whether

\angle TSU \sim \angle TRV

or not and explain your reasoning using

AA

SSS

SAS

Draw a graph of the line that contains the g has the given slope.

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(0,3): m=-2

Use the error formula to find

n

such that the error is less than or equal to

0.00001

using the Trapezoidal Rule.

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\int_{0}^{\frac{\pi}{2}}\sin x\,dx

11

mission is famous for putting the first man on the moon. Jenny informed her friends that the Apollo

11

lunar module was on the surface of the moon for

8 \times 10^7

milliseconds. What would be the most appropriate unit of time for Jenny to use instead of milliseconds?

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(i) Verify the Mean Value Theorem for

y=x^{3}-3 x+1

on the interval

[-2,2]

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(a) Verify that the hypotheses of the Mean value theorem holds.

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(b) Verify that the conclusion of the Mean value theorem holds.

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(ii) Illustrate (i-(b)) by a graph involving the extreme chord.

Find the values of

x

for which the given function is concave up, the values of

x

for which it is concave down, and any points of inflection.

y=-x^4+12x^3-12x+13

. What is/are the point(s) of inflection of

y

? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

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A. enter your response here (Type an ordered pair. Use a comma to separate answers as needed.)

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B. There are no points of inflection.

Use the Ratio Test to determine whether the series is convergent or divergent.

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\sum_{n=1}^{\infty}(-1)^{n-1} \frac{3^{n}}{2^{n} n^{3}}

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a_{n}

symmetry to sketch a graph

\qquad

of the function

y=36 x^{2}-100

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Label the intersection point for the axis of symmetry and the roots.

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- Add a new expression using the + in the upper left, then choose expression.

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- Type in a point you want to label using an ordered pair.

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(1,0)

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- Choose the Label checkbox.

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Once complete, choose the line that is the axis of symmetry.

3

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Write the complete factored form of the polynomial

f(x)

, given the indicated zero.

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f(x)=x^{3}-9 x^{2}+27 x-27

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3

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For the toolbar, press

A L T+F 10(P C)

A L T+F N+F 10

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DeltaMath Student

A 9

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Detamath Student A.

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A scale drawing of a

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2

[

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Find the equation that represents the proportional relationship in this graph, for

y

x

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17

Divide powers: variable bases

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Simplify. Express your answer

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\frac{10 j k^{3}}{2 j^{4} k}

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\square

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C_{2}

: Find the solution set by graphical method. (Find) ordered pairs.)

3 x-y=11 ; x-9=3 y

4

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y

6

less than the quotient of

x

4

. Which graph best shows this relationship?

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-o x.com/matn/grade-//area-ot-compound-figures-with-triangles

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What is the area of this figure?

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Write your answer using decimals, if necessary.

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Improve Your Math Fluency Series

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Instructions: Derive the indicated equation by combining trig identities (and, w) applicable, the definitions of secant, cosecant, and cotangent) together through substit nd applying algebra. There are no answers to these problems in the back of the ince the answer is instead given in the question).

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Use trig identities to show that

\cot ^{2} \theta+\csc ^{2} \theta=2 \csc ^{2} \theta-1

6

. Graph the function below. Then, ic discontinuities.

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f(x)=\left\{\begin{array}{cl} -\frac{1}{2} x-4 & \text { if } x<-1 \\ (x-2)^{2} & \text { if }-1 \leq x<3 \\ |x-5|-3 & \text { if } x \geq 3 \end{array}\right.

10

(C) Determine the quotient of divided by a polynomial of degree one a exceed the degree of the dividend (SUPF

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Y

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Which expression is equivalent to

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\left(x^{2}+3 x-32\right) \div(x-4) \text { ? }

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x+7-\frac{4}{x-4}

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x+8

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c x+7

7

g(x)=k^{2} x^{2}-10 k x+6 k+1

k

is a positive constant. The graph of

y=g(x)

(2,9)

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k

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x

-intercept(s) does the graph of

y=-g(x)+5

7

g(x)=k^{2} x^{2}-10 k x+6 k+1

k

is a positive constant. The graph of

y=g(x)

(2,9)

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k

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x

-intercept(s) does the graph of

y=-g(x)+5

4

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1 \mathrm{pts}

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You wish to test the following claim

\left(\mathrm{H}_{\mathrm{a}}\right)

at a significance level of

\alpha=0.05

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\begin{array}{l} \mathrm{H}_{\mathrm{o}}: p=0.42 \\ \mathrm{H}_{\mathrm{a}}: p>0.42 \end{array}

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You obtain a sample of size

n=143

in which there are

73

successful observations. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.

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What is the p-value? Round to

4

decimal places.

2

A solid has volume

7

cubic units. The equation

k=\sqrt[3]{\frac{V}{7}}

represents the scale factor of

k

by which the solid must be dilated to obtain an image with volume

V

cubic units. Select all points which are on the graph representing this equation.

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(0,0)

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(1,1)

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(1,7)

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(7,1)

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(14,2)

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\backslash((49,2) \backslash

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\backslash((56,2) \backslash)

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k

0

x

. Express your answer in simplest radical form if necessary.

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x^{2}=4

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x=

x

. Express your answer in simplest radical form if necessary.

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x^{2}=36

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x=

Express as a single fraction in its simplest form.

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1+y-\frac{y+2}{y-1}

-48=(v-6)^{2}

v

is a real number.

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Round your answer to the nearest hundredth.

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If there is more than one solution, separate them with commas.

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If there is no solution, click on

What are the solutions to the equation

(2 x+1)^{2}-(x+13)=3 x^{2}-2 x+2

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Enter your answors in the spaces provided Enter only your answers

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\begin{array}{l} x= \\ x= \end{array}

Solve the given equation. (Enter your answers as a comma-separated list. Let

k

be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)

7 \cos(\theta) \sin(\theta) + 4 \cos(\theta) = 0

Use the rational zeros theorem to determine the potential rational zeros of the polynomial function. Do not find the zeros.

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f(x)=x^{3}-6 x-81

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List the possible potential rational zeros of the polynomial function.

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\square

(Type an integer or a fraction. Use a comma to separate answers as needed.)

Use the rational zeros theorem to determine the potential rational zeros of the polynomial function. Do not find the zeros.

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f(x)=x^{3}-6 x-81

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List the possible potential rational zeros of the polynomial function.

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\square

If the product of the zeroes of the quadratic polynomial

p(x)=a x^{2}-6 x-6

4

, then find the value of

a

. Also, find the sum of the zeroes of the polynomial.

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P

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\frac{\alpha-1}{\alpha+1}\times\frac{\beta-1}{\beta+1}=\frac{1}{(\alpha+1)(\beta+1)}

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=\frac{\alpha \beta-\alpha-\beta+1}{\alpha \beta+\alpha+\beta+1}=\frac{\alpha \beta-(\alpha+\beta)+1}{\alpha \beta+(\alpha+\beta)+1}=\frac{3-2+1}{3+2+1}=\frac{2}{6}=\frac{1}{3}

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:. Required quadratic polynomial,

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p(x)=k{x^{2}-Sx+P}

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\Rightarrow p(x)=k{x^{2}-\frac{2}{3}x+\frac{1}{3}}

32

: If the product of the zeroes of the quadratic polynomial

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p(x)=ax^{2}-6x-6

4

, then find the value of

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a

. Also, find the sum of the zeroes of the polynomial.

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Solution : We have,

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*p(x)=ax^{2}-6x-6

. It is given that :

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\alpha \beta=4

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\frac{\alpha-1}{\alpha+1}\times\frac{\beta-1}{\beta+1}=\frac{1}{(\alpha+1)(\beta+1)}

0

be the zeroes of the polynomial.

If the product of the zeroes of the quadratic polynomial

p(x)=a x^{2}-6 x-6

4

, then find the value of

a

. Also, find the sum of the zeroes of the polynomial.

Find the value of

k

3 x^{2}+2 k x-k-5

has the sum of the zeroes as half of their product.

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f(x)=3 x^{3}-8 x^{2}-3 x+12

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g(x)=4 e^{(x+1)(x-3)}+2 x+2 .

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R

S

be the two regions enclosed by the graphs of

f

g

as shown in the graph.

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Find the sum of the areas of regions

R

S

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Use a graphing calculator and round your answer to three decimal places.

A=\left[\begin{array}{rr}2 & 3 \\ -2 & -5\end{array}\right]

B=\left[\begin{array}{rr}-1 & 8 \\ 6 & -7\end{array}\right]

. Find a matrix

X

satisfying the given equation.

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5 X=2 A+3 B

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\square

(Simplify your answer.)

Use the quadratic formula to solve. Express your answer in simplest form.

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a^{2}+4 a+4=0

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1

2

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(†) Additional Solution

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\Theta

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a=

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\square

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16

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f

f(x)=a x^{2}+b x+c

a, b

c

are constants and

1<a<4

y=f(x)

x y

-plane passes through points.

(11,0)

(-2,0)

a

is an integer, what could be the value of

f(x)=a x^{2}+b x+c

0

p(x)=\frac{-3 x^{2}+5 x-1}{x^{2}+3}

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(a) Identify the horizontal asymptotes (if any).

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(b) If the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s).

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Separate multiple equations of asymptotes with commas as necessary. Select

p=(w-30)\left(w^{2}+178 w+792\right.

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-89

is a double zero of the polynomial equation, which of the following could be the graph of the equation in the

w p

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1

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p

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tion graphs - Harder example

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er nonlinear graphs: medium

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other nonlinear graphs | Lesson

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tors and graphs - Basic example

A B C D

is a rhombus with diagonals that intersect at

F

. Find the value of '

x

m \angle A B D=(2 x+3)^{\circ}

m \angle D B C=(4 x-1)^{\circ}

A=\int_{0}^{1}\cos x\,dx

A

L

R

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T

approximations with

n=100

subintervals. Which is true?

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L < A < T < R

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L < T < A < R

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R < A < T < L

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R < T < A < L

y=12(0.25)^{x}+1

is shown in the

x y

-plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?

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1

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x

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y

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y

x

becomes very large

f(x)=\left\{\begin{array}{cl} -x^{2}+3 x & \text { if } x<1 \\ 2 x+2 & \text { if } x \geq 1 \end{array}\right.

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evaluate the limit

\lim _{x \rightarrow 1} f(x)

. If the limit does not exist, write DNE.

Google Classroom

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Rewrite the expression in the form

b^{n}

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\frac{b^{-2}}{b^{4}}=\square

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Stuck? Review related articles/videos or use a hint.

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