Please answer the following multiple choice questions. Click on your answer. Please feel free to use...

Please answer the following Please answer the following multiple choice questions. multiple choice questions.

Click on your answer. Click on your answer.

Please feel free to use paper or Please feel free to use paper or graphing calculators to find the graphing calculators to find the

correct answer.correct answer.

Click here to go to question #1Click here to go to question #1

When finding solutions to When finding solutions to equations, you are finding theequations, you are finding the

A) x-interceptsA) x-intercepts

B) y-interceptsB) y-intercepts

C) vertex pointsC) vertex points

D) line of symmetryD) line of symmetry

Correct!Correct!

Great job!Great job!

Click here to go to tClick here to go to the next questionhe next question

OOPSOOPS

Please try again.Please try again.

Click here to return tClick here to return to Question #1o Question #1

How many solutions does the How many solutions does the equation have?equation have?

10x10x3 3 + 20x+ 20x22 + x + 2 = 0 + x + 2 = 0

A) 6A) 6

B) 5B) 5

C) 3C) 3

D) 2D) 2

Right!Right!

You knew to look for the highest You knew to look for the highest degree of exponent.degree of exponent.

Click here to go to quClick here to go to question #3estion #3

Sorry, that’s incorrect. Sorry, that’s incorrect.

Don’t add the exponents, but the Don’t add the exponents, but the answer does deal with exponentsanswer does deal with exponents

Click here to go bacClick here to go back to #2k to #2

OOPSOOPS

Try again, but think about what Try again, but think about what the exponents tell us.the exponents tell us.

Click here to go baClick here to go back to #2ck to #2

Do all of the solutions to these Do all of the solutions to these problems have to be real problems have to be real

number solutions?number solutions?

A) YesA) Yes

B) NoB) No

Correct!Correct!

They can have a combination of They can have a combination of real and imaginary solutions!real and imaginary solutions!


That’s incorrect.That’s incorrect.

Remember they can have a Remember they can have a combination of real and imaginary combination of real and imaginary

solutions.solutions.


If factoring by grouping, which If factoring by grouping, which terms would you group together to terms would you group together to

factorfactor

xx33 + x + x22 + x + 1 + x + 1

A) (xA) (x33 + x + x22) + (x + 1)) + (x + 1)

B) (xB) (x33 + x) + (x + x) + (x22 + 1) + 1)

C) (xC) (x33 + 1) + (x + 1) + (x22 + x) + x)

D) A, B, & C workD) A, B, & C work

Super!Super!

Although they all work, choices A Although they all work, choices A and B make this problem much and B make this problem much

easier to solve.easier to solve.

Click here to go toClick here to go to question #5 question #5

This is a correct way to group, but This is a correct way to group, but they actually all work. they actually all work.

Choices A) (xChoices A) (x33 + x + x22) + (x + 1)) + (x + 1) B) (x B) (x33 + x) + (x + x) + (x22 + 1) + 1) make this problem much easier to make this problem much easier to

solve than choicesolve than choice C) (x C) (x33 + 1) + (x + 1) + (x22 + x) + x)

Click here to go tClick here to go to question #5o question #5

If you chose to group like thisIf you chose to group like this

(x (x33 + x + x22) + (x + 1) ) + (x + 1)

What would be the next steps?What would be the next steps?

A) x(xA) x(x22 + x) + (1 + x) + x) + (1 + x) B) xB) x22(x + 1) + 1(x + 1)(x + 1) + 1(x + 1)

(x)(x(x)(x22 + x)(2) + x)(2) (x(x22 + 1)(x + 1) + 1)(x + 1)

C) xC) x33(1 + x) + 1(x + 1)(1 + x) + 1(x + 1) D) xD) x22(x) + (x + 1)(x) + (x + 1)

(x(x33 + 1)(x + 1) + 1)(x + 1) x(xx(x22 + 1) + 1)

CorrectCorrect


Click here to go to Click here to go to question #6question #6

OOPSOOPS

Please look at all the choices Please look at all the choices and try again.and try again.

Click here to go back Click here to go back to the problemto the problem

Now we have the factored formNow we have the factored form

(x(x22 + 1)(x + 1) + 1)(x + 1)

Are all of the solutions real Are all of the solutions real numbers?numbers?

A) YesA) Yes

B) NoB) No

That’s correct.That’s correct.

(x(x22 + 1) gives imaginary solutions + 1) gives imaginary solutions&&

(x + 1) gives a real solution(x + 1) gives a real solution

Click here to go onClick here to go on


(x(x22 + 1) gives imaginary solutions + 1) gives imaginary solutions&&

(x + 1) gives a real solution(x + 1) gives a real solution


What are the real solutions to What are the real solutions to

(x(x22 + 1)(x + 1) = 0 + 1)(x + 1) = 0 ? ?

____

A) 1, -1A) 1, -1 B) √1, -1B) √1, -1

____

C) ±√1C) ±√1 D) -1D) -1

That’s correct!That’s correct!

-1 is the only real solution-1 is the only real solution


Sorry, that’s not right.Sorry, that’s not right.

Remember to find solutions set Remember to find solutions set each factor equal to zero and each factor equal to zero and

solve.solve.

Click here to go bacClick here to go back to the problemk to the problem

Which is the correct graph forWhich is the correct graph for x x33 + x + x22 + x + 1 + x + 1

A)A) B)B)

C)C) D)D)


You must have remembered what You must have remembered what graphs with a 3graphs with a 3rd rd degree polynomial degree polynomial

look like, then you only had to look like, then you only had to remember the real solution to figure remember the real solution to figure

it out.it out.OrOr

You used your graphing calculator.You used your graphing calculator. Click here to go to queClick here to go to que

stion #9stion #9

You’re closeYou’re closebut remember what the real but remember what the real solution was to this problem.solution was to this problem.

We solved it in the last question.We solved it in the last question.

Click here to go bacClick here to go back to graphsk to graphs


Think about what shape the graph Think about what shape the graph of a 3of a 3rdrd degree polynomial should degree polynomial should

look like.look like.

Click here to go bacClick here to go back to graphsk to graphs

How many solutions does the How many solutions does the equation have?equation have?

4x4x66 – 20x – 20x44 = -24x = -24x22

A) 12A) 12

B) 6B) 6

C) 4C) 4

D) 1D) 1

Great!Great!

You’re thinking now.You’re thinking now.


No, remember you don’t add the No, remember you don’t add the exponents. What do the exponents. What do the

exponents tell us?exponents tell us?

Click here to go back tClick here to go back to the problemo the problem

Sorry that’s incorrect.Sorry that’s incorrect.

What do the exponents tell us?What do the exponents tell us?

Click here to go back toClick here to go back to the problem the problem

Factor the followingFactor the following

xx33 - 8 - 8

Click here when you hClick here when you have an answerave an answer

Select the answer that you found.Select the answer that you found.

If your answer is not below, click below to return to If your answer is not below, click below to return to the problemthe problem

A) (xA) (x22 + 4)(x – 2) + 4)(x – 2)

B) (x – 2)(xB) (x – 2)(x22 + 2x + 4) + 2x + 4)

C) (x + 2)(xC) (x + 2)(x22 + 2x + 4) + 2x + 4)

D) (x – 2)(xD) (x – 2)(x22 – 2x – 4) – 2x – 4)

Go back to problemGo back to problem

Super!Super!

You’re doing great and you’re You’re doing great and you’re half way finished.half way finished.

Click here to go tClick here to go to question #11o question #11

No, think about the rules for No, think about the rules for factoring the difference of factoring the difference of

cubes.cubes.

Click here to go bClick here to go back to choicesack to choices

Click here to see the problem again

Not quite, but you’re close.Not quite, but you’re close.

Think about the signs that you Think about the signs that you need when factoring the need when factoring the

difference of cubes.difference of cubes.

Click here to go bacClick here to go back to the choicesk to the choicesClick here to go back to

the problem

Which of the following are possible Which of the following are possible solutions to the functionsolutions to the function

xx33 + 3x + 3x22 + 10x = -30 + 10x = -30

A) 3A) 3

B) -3B) -3

C) 10C) 10

D) 5D) 5


Did you remember that there are several Did you remember that there are several ways to get this solution.ways to get this solution.

If you factored and then set them equal to If you factored and then set them equal to zero, you could have checked your answer zero, you could have checked your answer by graphing the equation, and finding the by graphing the equation, and finding the

x-intercepts. x-intercepts.

(If you haven’t graphed it try now.)(If you haven’t graphed it try now.) Click here to go to tClick here to go to t

he next problemhe next problem

No, please try again.No, please try again.

Hint 1: set the equation equal to zeroHint 1: set the equation equal to zero

Hint 2: factor, set each equal to zero Hint 2: factor, set each equal to zero and solveand solve

Hint 3: graph and look at the Hint 3: graph and look at the x-interceptsx-intercepts


Find one possible solution to the Find one possible solution to the equation belowequation below

-2x-2x33 – 4x – 4x22 – 3x – 6 = 0 – 3x – 6 = 0

A) 3/2A) 3/2

B) -3B) -3

C) -2C) -2

D) No real solutionsD) No real solutions

Super job!Super job!

You’re doing great!You’re doing great!



Hint 1: factor, set each equal to Hint 1: factor, set each equal to zero and solvezero and solve

Hint 2: graph and look at the Hint 2: graph and look at the x-interceptsx-intercepts


Which could be the graphical Which could be the graphical representation ofrepresentation of

32x32x33 - 4 - 4

A)A) B) B)

CC)) D) D)

Correct!Correct!



No, think about what No, think about what shape the graph of a cubic shape the graph of a cubic

should look like.should look like.

Click here to go bacClick here to go back to the graphsk to the graphs

No, but you have the right No, but you have the right shape.shape.

Click here to go Click here to go back to the graphsback to the graphs

FactorFactor

27x27x33 + 216 + 216

Click here when you Click here when you have an answerhave an answer

Select your answer, if it is not Select your answer, if it is not here click here to go back to the here click here to go back to the problem so you may try againproblem so you may try again

A) (3x – 2)(9xA) (3x – 2)(9x22 – 2x + 4) – 2x + 4)

B) 27(x + 2)(xB) 27(x + 2)(x22 – 2x + 4) – 2x + 4)

C) 27(x – 2)(xC) 27(x – 2)(x22 + 2x – 4) + 2x – 4)

D) (3xD) (3x22 – 6)(3x +6) – 6)(3x +6)


You remembered everything, You remembered everything, you must be thinking.you must be thinking.



Taking out the GCF might be a Taking out the GCF might be a good start.good start.

Click here to go Click here to go back to answersback to answers

Click here to go back to the problem

You remembered to take out the You remembered to take out the GCF, but check your signs.GCF, but check your signs.

Click here to go Click here to go back to answersback to answers

Click here to go back to the problem

Which of the following is a solution Which of the following is a solution to the functionto the function

2x2x33 + 54 = 0 + 54 = 0

A) -3A) -3

B) 2B) 2

C) 3C) 3

D) 9D) 9

Correct, great job.Correct, great job.

There were several ways to come up with this There were several ways to come up with this answer. answer.

Here are a few of the ways:Here are a few of the ways:- graph, look for x-intercepts- graph, look for x-intercepts- solve algebraically- solve algebraically- because it’s multiple choice you could have - because it’s multiple choice you could have used substitution and tried all possible used substitution and tried all possible answers. answers.


OOPS, that’s wrong.OOPS, that’s wrong.

Think about possible ways to solve this Think about possible ways to solve this equation:equation:

Here are a few of the ways:Here are a few of the ways:- graph, look for x-intercepts- graph, look for x-intercepts- solve algebraically- solve algebraically- because it’s multiple choice you could have - because it’s multiple choice you could have used substitution and tried all possible used substitution and tried all possible

answers. answers. Click here to go Click here to go back to the problemback to the problem

You’re very close, go back and You’re very close, go back and check your work.check your work.

You may also want to try it another way, such as:You may also want to try it another way, such as:

- graph, look for x-intercepts- graph, look for x-intercepts- solve algebraically- solve algebraically- because it’s multiple choice you could have - because it’s multiple choice you could have used substitution and tried all possible used substitution and tried all possible answers. answers.

Click here to go back Click here to go back to the problemto the problem

How many solutions does the How many solutions does the following equation have?following equation have?

2x2x55 + 24x = 14x + 24x = 14x33

A) 9A) 9

B) 5B) 5

C) 3C) 3

D) 2D) 2


You knew it was the largest You knew it was the largest exponent. Did you also remember exponent. Did you also remember

that not all solutions have to be real that not all solutions have to be real solutions?solutions?

Click to see part 2 Click to see part 2 of this problemof this problem

Sorry, that’s incorrect. Sorry, that’s incorrect.

Don’t add the exponents, but Don’t add the exponents, but the answer does deal with the answer does deal with

exponentsexponents

Click here to go back Click here to go back to the equationto the equation

OOPSOOPS

Try again, but think about what Try again, but think about what the exponents tell us.the exponents tell us.

Click here to go Click here to go back to the equationback to the equation

What are possible solutions toWhat are possible solutions to

2x 2x55 + 24x = 14x + 24x = 14x33

A) 0, 2, -2, 3, -3A) 0, 2, -2, 3, -3

B) 0, 3, 4B) 0, 3, 4 __ ____ __

C) 0, -2, 2, √3, -√3C) 0, -2, 2, √3, -√3

D) -2, 2D) -2, 2

Click here to go to Click here to go to the last questionthe last question

Click here to go back and see the original equation

Super!Super!

Did you solve this algebraically, Did you solve this algebraically, graphically, or by using the solutions graphically, or by using the solutions

that were given?that were given?

If you only tried one method, try If you only tried one method, try another before you move on.another before you move on.


Hint: Check your math or try to Hint: Check your math or try to solve the problem another way solve the problem another way

(graphically).(graphically).

Click here to go back Click here to go back to the equationto the equation

What are possible solutions toWhat are possible solutions to

2x2x33 – 5x – 5x22 + 18x = 45 + 18x = 45

A) 2.5A) 2.5

B) 2.5, 3, -3B) 2.5, 3, -3

C) 2.5, -9C) 2.5, -9

D) No real solutionsD) No real solutions


There is only 1 real solution, the There is only 1 real solution, the other 2 are imaginary. other 2 are imaginary.

Click here, pleaseClick here, please

No, think about all the ways that we No, think about all the ways that we can find the solution.can find the solution.

Then try again, remember not all Then try again, remember not all solutions must be real.solutions must be real.

If you haven’t tried graphing click this If you haven’t tried graphing click this link to try that method or go back to the link to try that method or go back to the

problem and try another method.problem and try another method.

Click here to go Click here to go back to the problemback to the problem

We’re all through, I hope you We’re all through, I hope you have a better understanding of have a better understanding of

solving equations.solving equations.

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