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MA 111 / 115, section WPU F20 Waters, Fall 2020

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WebAssign: §1.2 (Homework)

Steve Waters

Warner Paci c College, OR

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QUESTION

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10

11

POINTS

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TOTAL SCORE

–/18

0.0%

Due Date

FRI, SEP 11, 2020

11:30 AM PDT

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Assignment Submission & Scoring

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining

for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

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[–/3 Points]

MY NOTES

LARCAABLS2 1.2.016.

DETAILS

ASK YOUR TEACHER

PRACTICE ANOTHER

One positive number is one-sixth of another number. The difference of the two numbers is 75. Find the

numbers.

(a) Use the verbal description to write a verbal model.

difference = (another number) − (one number)

difference = (one number) + (another number)

difference = −(another number) − (one number)

difference = (another number) + (one number)

difference = −(one number) − (another number)

(b) Assign labels to the quantities in the verbal model.

difference

= 75

difference

= x

—Select—

=

1

x

6

(c) Use the labels to write a mathematical model.

(d) Solve the problem. (Enter your answers as a comma-separated list.)

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[–/3 Points]

MY NOTES

DETAILS

LARCAABLS2 1.2.020.

ASK YOUR TEACHER

PRACTICE ANOTHER

The price of a shirt has been discounted by $10. The sale price is $24.95. What percent of the original

list price is the discount?

(a) Use the verbal description to write a verbal model.

change in price = (percent) − (original price)

change in price = (percent)/(original price)

change in price = (original price)/(percent)

change in price = (original price) + (percent)

change in price = (percent)(original price)

(b) Assign labels to the quantities in the verbal model.

—Select—

= 10

—Select—

= r

—Select—

= 34.95

(c) Use the labels to write a mathematical model.

(d) Solve the problem. (Round your answer to the nearest whole number.)

r=

%

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WebAssign: §1.2 – MA 111 / 115, section WPU F20 Waters, Fall 2020 | WebAssign

[–/1 Points]

DETAILS

MY NOTES

LARCAABLS2 1.2.030.

ASK YOUR TEACHER

PRACTICE ANOTHER

Use the following information. Restaurants tend to serve food in larger portions now than they have in

the past. Several examples are shown in the table. Find the percent increase in size from the past to

2011 for the indicated food item. (Round your answer to one decimal place.)

Small French fries

%

Food or drink item Past size 2011 size

4.

Small soft drink

8 fl oz

15 fl oz

Small French fries

2.3 oz

2.4 oz

Large French fries

3.6 oz

5.2 oz

Pizza

9 in.

14 in.

[–/3 Points]

MY NOTES

DETAILS

LARCAABLS2 1.2.036.

ASK YOUR TEACHER

PRACTICE ANOTHER

You accept a new job with a starting salary of $51,000. You receive a 5% raise at the start of your

second year, a 5.6% raise at the start of your third year, and an 11.4% raise at the start of your fourth

year. (Round your answers to two decimal places.)

(a) Find your salary for the second year.

$

(b) Find your salary for the third year.

$

(c) Find your salary for the fourth year.

$

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5.

[–/2 Points]

MY NOTES

WebAssign: §1.2 – MA 111 / 115, section WPU F20 Waters, Fall 2020 | WebAssign

DETAILS

LARCAABLS2 1.2.039.

ASK YOUR TEACHER

PRACTICE ANOTHER

A picture frame (see figure) has a total perimeter of 4 feet. The width of the frame is 0.61 times its

length. Find the dimensions of the frame. (Round your answers to two decimal places.)

6.

width

ft

length

ft

[–/1 Points]

MY NOTES

DETAILS

LARCAABLS2 1.2.044.MI.

ASK YOUR TEACHER

PRACTICE ANOTHER

To get an A in a course, you need an average of 90% or better on four tests. The first three tests are

worth 100 points each and the fourth is worth 200 points. Your scores on the first three tests are 87, 89,

and 85. What must you score on the fourth test to get an A for the course?

points (or greater)

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WebAssign: §1.2 – MA 111 / 115, section WPU F20 Waters, Fall 2020 | WebAssign

[–/1 Points]

MY NOTES

DETAILS

LARCAABLS2 1.2.054.

ASK YOUR TEACHER

PRACTICE ANOTHER

To determine the height of a building, you measure the building’s shadow and the shadow of a four-foot

1

stake, as shown in the figure. How tall is the building if x = 80 ft, a = 4 ft, and b = 3

ft? (Round your

2

answer to two decimal places.)

ft

8.

[–/1 Points]

MY NOTES

DETAILS

LARCAABLS2 1.2.052.

ASK YOUR TEACHER

PRACTICE ANOTHER

Students are traveling in two cars to a football game 140 miles away. One car travels at an average

1

speed of 45 miles per hour. The second car starts

hour later and travels at an average speed of 55

2

miles per hour. How long will it take the second car to catch up to the first car?

hr

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[–/1 Points]

9.

WebAssign: §1.2 – MA 111 / 115, section WPU F20 Waters, Fall 2020 | WebAssign

DETAILS

LARCAABLS2 1.2.064.

MY NOTES

ASK YOUR TEACHER

MY NOTES

ASK YOUR TEACHER

Solve for the indicated variable.

Volume of a Rectangular Prism

Solve for l in V = lwh.

l=

10.

[–/1 Points]

DETAILS

LARCAABLS2 1.2.067.

Solve for the indicated variable.

Discount

Solve for L in S = L − RL.

L=

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[–/1 Points]

MY NOTES

DETAILS

LARCAABLS2 1.2.072.

ASK YOUR TEACHER

PRACTICE ANOTHER

A trough is x = 12 feet long, y = 3 feet deep, and y = 3 feet wide (see figure). Find the depth of the

water when the trough contains 70 gallons of water. (1 gallon ≈ 0.13368 cubic foot. Round your answer

to two decimal places.)

ft

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F20 MAT 101S Assignment E

Factoring Polynomial Functions (Content from Unit 4 in MAT 101)

Zeros

Factors

𝑥-values that make

the function equal

0

𝑥-values of the

𝑥-intercepts

How do zeros & factors relate?

If 2 is a zero, then (𝑥 − 2) is a

factor.

If -1 is a zero, then (𝑥 + 1) is a

factor.

two expressions that multiply together

any polynomial that divides evenly into

𝑓(𝑥)

of the form (𝑥 − “zero of polynomial”)

of the form (𝑥 − 𝑐)

Example – Factor: 𝒇(𝒙) = 𝟐𝒙𝟑 − 𝟓𝒙𝟐 − 𝒙 + 𝟔

*Notice this is in function notation.

Step 1: Find all possible zeros of the function. To narrow our search for zeros, we can use information from

the function to figure out numbers that could possibly be a zero.

P is the constant in the function. In our case, P is 6.

Q is the leading coefficient. In our case, Q is 2.

List all factors of P: 1, 2, 3, 6.

(What multiplied by what makes 6? Use all of those numbers.)

List all factors of Q: 1, 2.

(What multiplied by what makes 2? Use all of those numbers.)

Use

𝑃

𝑄

to find all possible zeros. Remember the zero could possibly be negative or positive.

𝑃 ±1, ±2, ±3, ±6

1 2 3 6 1 2 3 6

=

= ± ,± ,± ,± ,± ,± ,± ,±

𝑄

±1. ±2

1 1 1 1 2 2 2 2

1

3

Simplify = ±1, ±2, ±3, ±6, ± 2 , ±1, ± 2 , ±3

𝟏

𝟑

Omit Duplicates = ± 𝟐 , ±𝟏, ± 𝟐 , ±𝟐, ±𝟑, ±𝟔

So, this list gives us a list of all POSSIBLE zeros of the function above.

How do we find the ACTUAL zeros?

Step 2: Use synthetic division to find a number that gives a ZERO remainder.

Let’s start with one of the possible zeros in the list. It doesn’t matter which one you pick to try first.

Use the process for synthetic division previously taught.

Let’s try 1 as a possible zero.

1

2

-5

-1

2

-3

2

-3

-4

6

-4

2

* So, our remainder was 2, which is not zero.

Therefore, 1 is not a zero of 𝑓(𝑥).

Let’s try -1 as a possible zero.

-1

2

-5

-2

2

-7

-1

7

6

6

-6

0

Bingo! Our remainder was zero.

Therefore, −1 is a zero of f(x), and (𝑥 + 1) is a factor of 𝑓(𝑥).

Step 3: Rewrite the equation with the new factor.

Let’s rewrite 𝑓(𝑥) with our new factor. The format will be (𝑥 – “𝑧𝑒𝑟𝑜”)(𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡).

𝑓 (𝑥) = (𝑥 + 1)(2𝑥 2 − 7𝑥 + 6)

Step 4: Using a method of factoring to factor the trinomial factor. Now that we have a semi-factored form,

let’s see what else we can do:

𝑓 (𝑥) = (𝑥 + 1)(2𝑥 2 − 7𝑥 + 6)

2𝑥 2 − 7𝑥 + 6 can be factored even further into (2𝑥 − 3)(𝑥 − 2).

*Use grouping to factor.

So, the final factored form is: 𝒇(𝒙) = (𝒙 + 𝟏)(𝟐𝒙 − 𝟑)(𝒙 − 𝟐).

𝟑

If we set each of these factors = 0, we get the zeros or solutions to 𝒇(𝒙) = 𝟎: −𝟏, 𝟐 , 𝟐.

F20 MAT 101S Assignment E

Name: ________________________________

1a. [2 pts] Show that 2 is a zero of 𝒇(𝒙) = 𝟒𝒙𝟑 − 𝟏𝟐𝒙𝟐 + 𝟓𝒙 + 𝟔 using synthetic division.

1b. [2 pts] Write 𝑓 (𝑥) in the form (𝑥 − 𝑐)(𝑄(𝑥)) : _________________________________________________________________

1c. [3 pts] Factor 𝑄(𝑥) using any method you choose.

Factored form of 𝑄(𝑥): ___________________________________________________________________________________________

1d. [3 pts] Write the factored form of 𝑓(𝑥). 𝑓(𝑥) =_____________________________________________________________

Definition: A factor of an integer is any number that divides that integer evenly (zero remainder).

2a. [2pts] List the factors of 15: _____________________________________________

2b. [2pts] List the factors of 20: ______________________________________________

2c. [2pts] List the factors of 24: _______________________________________________

2d. [4 pts] If you wanted to create all the possible fractions you could given 1, 2, 5, and 10 as a list of

numbers you must use for the numerator and 1, 2, and 4 as a list of numbers you must use for the

denominator, what would be your list of fractions that would be created using these numbers.

1

2

1

1

I’ll get you started: ,

,

5

1

,

10

1

1

, , __________________________________________________________________________________

2

Then, simply each fraction. This would give:

1

1, 2, 5, 10, 2, __________________________________________________________________________

3. Follow the step-by-step guide below to help you write the following function in factored form. Then,

solve 𝒇(𝒙) = 𝟎.

𝑓 (𝑥) = 2𝑥 3 − 3𝑥 2 − 3𝑥 + 2

[2 pts] Part A: Find all the possible zeros.

What is P? ______

What is Q? ______

𝑃

What is 𝑄? Be sure to simplify.

Write the list of all possible zeros: _____________________________________________

[2 pts] Part B: Find one actual zero.

Use synthetic division to test the possible zeros. How do you know if it is an actual zero?

[2 pts] Part C: Rewrite the function by turning the actual zero into a factor and the coefficients into a

trinomial factor.

Rewrite the function with the new factor: 𝑓(𝑥) = _______________________________

[2 pts] Part D: Factor the trinomial factor (the quotient from above).

Factor the trinomial factor in the function above:

[3 pts] Part E: Write the final factored form of the function. Remember: 𝒇(𝒙) = (𝒙 − 𝒛𝒆𝒓𝒐)(factored quotient).

Final factored form: 𝑓 (𝑥) =_________________________________

[3 pts] Part F: Set each of the binomial factors equal to zero and solve each for 𝒙 to find the solutions to

𝒇(𝒙) = 𝟎.

What are the solutions to 𝑓 (𝑥) = 0? _________________________________________

4. [15 pts] Write the following function in factored form. Then, solve 𝒇(𝒙) = 𝟎.

Write your final answers at the bottom.

𝑓 (𝑥) = 2𝑥 3 − 3𝑥 2 − 39𝑥 + 20

Answers:

Write the list of all possible zeros: _______________________________________________________________

Final factored form: 𝒇(𝒙) =_________________________________________________________________________

What are the solutions to 𝒇(𝒙) = 𝟎? _______________________________________________________________

5. [15 pts] Write the following function in factored form. Then, solve 𝒇(𝒙) = 𝟎.

Write your final answers at the bottom.

𝑓 (𝑥) = 2𝑥 3 − 𝑥 2 − 22𝑥 − 24

Answers:

Write the list of all possible zeros: ________________________________________________________________

Final factored form: 𝒇(𝒙) =___________________________________________________________________________

What are the solutions to 𝒇(𝒙) = 𝟎? _______________________________________________________________

6. Find the LCM of each set of terms.

a. [2 pts] 90 and 20

LCM: ____________________________________________________

b. [3 pts] 4𝑥 2 , 2𝑥, & 11

LCM: _________________________________________________________

c. [3 pts] 𝑥 + 5 and 𝑥 − 5.

LCM: ____________________________________________________________

Definition: The least common denominator (LCD) of two or more fractions is the least common multiple

(LCM) of the denominators.

2

3

Example: Find the LCD of and .

7

5

Solution: We are looking for the LCM of the two denominators: 7 & 5. So, the answer is 𝟑𝟓.

7. Find the LCD of each set of rational expressions.

1

7

a. [2 pts] 2 and 11

1

c. [3 pts] 𝑥 ,

2

𝑥+4

, and

1

𝑥 2+4𝑥

𝑥

7

b. [3 pts] 𝑥+2 and 𝑥−2

Solving a RATIONAL EQUATION: Use this as a reference for the next two problems.

2

𝑥+4

1

17

+ 𝑥−4 = 𝑥 2−16

Step 1: Factor all denominators.

2

𝑥+4

1

17

+ 𝑥−4 = (𝑥+4)(𝑥−4)

The LCD is (𝑥 + 4)(𝑥 − 4).

Step 2: Find the LCD.

Step 3: Multiply each term by a factor equaling 1 to get a common denominator.

2

(𝑥 − 4)

1

(𝑥 + 4)

17

∙

+

∙

=

𝑥 + 4 (𝑥 − 4) 𝑥 − 4 (𝑥 + 4) (𝑥 + 4)(𝑥 − 4)

Step 4: Multiply each term by the LCD:

(𝑥 + 4)(𝑥 − 4) (

(𝑥 − 4)

(𝑥 + 4)

2

1

17

∙

) + (𝑥 + 4)(𝑥 − 4) (

∙

) = (𝑥 + 4)(𝑥 − 4) (

)

𝑥 + 4 (𝑥 − 4)

𝑥 − 4 (𝑥 + 4)

(𝑥 + 4)(𝑥 − 4)

Step 5: Cancel out common factor, eliminating the fractions.

2

(𝑥−4)

1

(𝑥+4)

17

(𝑥 + 4)(𝑥 − 4) (𝑥+4 ∙ (𝑥−4)) + (𝑥 + 4)(𝑥 − 4) (𝑥−4 ∙ (𝑥+4)) = (𝑥 + 4)(𝑥 − 4) ((𝑥+4)(𝑥−4))

Step 6: Write the remaining terms. 2(𝑥 − 4) + 1(𝑥 + 4) = 17

Step 7: Solve the equation remaining. (This one is not quadratic, but if it is, you would have to factor or use

other methods to solve.)

2𝑥 − 8 + 𝑥 + 4 = 17

Linear Equation: Solve for 𝑥.

3𝑥 − 4 = 17

+4

+4

3𝑥 = 21

3

3

𝑥=7

Step 8: Check your answer! Will it make the denominator equal zero?

The LCD is (𝑥 + 4)(𝑥 − 4).

Plug the possible solution(s) into the LCD & see if it makes one of the factors =0.

Let’s test 𝑥 = 7 in the LCD: (7 + 4)(7 − 4) = (11)(3) = 33 which is NOT zero, so 7 works and is a solution!

Therefore, we can use 7 as our answer.

Final Solution: 𝑥 = 7

*Note, some equations are quadratics after eliminating the fractions. Factor and solve. Check EACH possible

answer in the LCD to see if it makes it = 0. If one does and one does not, eliminate the one that makes the

LCD =0. You would only have one solution. If BOTH make the LCD = 0, your answer is NO SOLUTION

(DNE).

8. [10 pts] Solve the following rational equation using the previous reference page as a guide.

2

𝑥+3

5

37

+ 𝑥−3 = 𝑥 2−9

9. [10 points] Solve.

1

3

9

+ 𝑥+3 = 2𝑥+6

5

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