When expressions of the form (x −r)(x − s) are multiplied out, a quadratic polynomial is obtained. For instance, (x −2)(x −(−7))= (x −2)(x + 7) = x2 + 5x − 14.
a. What can be said about the coefficients of the polynomial obtained by multiplying out (x −r)(x − s) when both r and s are odd integers? when both r and s are even integers? when one of r and s is even and the other is odd?
b. It follows from part (a) that x2 − 1253x + 255 cannot be written as a product of two polynomials with integer coefficients. Explain why this is so.

The area of the regular polygon is 362.5 units².

The polygon above is a pentagon because it has five sides. The polygon is a regular polygon because all its sides are equal to each other.

The side length and apothem of the pentagon is given. Therefore, the area of the regular pentagon can be found as follows:

area of the regular pentagon = 1 / 2 × perimeter × apothem

Therefore,

perimeter of the regular pentagon = 14.5 × 5 = 72.5 units

apothem = 10 units

Hence,

area of the regular pentagon =  1 / 2 × 72.5 × 10

area of the regular pentagon = 72.5 × 5

area of the regular pentagon = 362.5 units²

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Answer:

  $39.45, about 31%

Step-by-step explanation:

You want to know the markdown represented by a sale price of $88.54 on a regular price of $127.99.

The dollar amount of the markdown is ...

  88.54 -127.99 = -39.45

The price was marked down $39.45.

The percentage markdown from the original price is ...

  -39.45/127.99 × 100% ≈ -30.823% ≈ -31%

The original price was marked down about 31% to get the sale price.

__

Additional comment

The negative price change means the price was marked down. If the change were positive, it would signify a markup.

Answers:

Nora = $40

Sara = $55

=========================================================

Work Shown:

n = amount Nora earned

s = amount Sara earned

n+s = 95 .... total amount earned

s = n+15 .... since Sara earned $15 more than Nora

--------------

I'll use the substitution method to solve.

n+s = 95

n+n+15 = 95 ....... replaced "s" with n+15

2n+15 = 95

2n = 95-15

2n = 80

n = 80/2

n = 40 dollars is the amount Nora earned

s = n+15

s = 40+15

s = 55 dollars is the amount Sara earned

--------------

Check:

Nora+Sara = 40+55 = 95 dollars total earned

This confirms the answer. Also, we can see that Sara earned $15 more than Nora to help further confirm the answer.

Answer:

True.

Step-by-step explanation:

Alternate interior angles are congruent, meaning they have equal measure. When we have two parallel lines that are intersected by a transversal, and again my parallel lines are identified by using the same number of arrows, then two special angles are congruent and that is alternate interior angles.

Answer: A positive whole number multiplied by any whole number will remain positive. In the case of the squares of 4, it will always end in a 6 which is a positive number.

Step-by-step explanation:

4^2= 16

16^2 = 256

256^2= 65,536

etc.

Check the picture below.

[tex]\cot(\gamma )=\cfrac{\stackrel{adjacent}{24}}{\underset{opposite}{18}}\implies \cot(\gamma )= \cfrac{4}{3}[/tex]

The solutions to the system of equations are (x,y) = (0,1) and (5/2,3)

What is Algebraic Equations?

An algebraic equation is also known as a polynomial equation because both sides of the equal sign contain polynomials. An algebraic equation is built up of variables, coefficients, constants as well as algebraic operations such as addition, subtraction, multiplication, division, exponentiation, etc.

To solve the system of equations algebraically, we can set the two expressions for y equal to each other and solve for x:

[tex]-2x + 1 = -2x^2 + 3x + 1[/tex]

Simplifying, we get:

[tex]-2x^2 + 3x + 1 + 2x - 1 = 0[/tex]

[tex]-2x^2 + 5x = 0[/tex]

Factorizing x, we get:

x(-2x + 5) = 0

Therefore, the solutions for x are:

x = 0 or x = 5/2

To find the corresponding values of y, we can substitute these values of x into either of the original equations for y:

For x = 0:

y = -2(0) + 1 = 1

For x = 5/2:

[tex]y = -2(5/2)^2 + 3(5/2) + 1 = -13/2 + 15/2 + 1 = 3[/tex]

Therefore, the solutions to the system of equations are:

(x,y) = (0,1) and (5/2,3)

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1. To break down 12x² into its prime factors,Start dividing 12 by 2, which gives you 6. Then, break 6 by 2, which gives 3. To write 12x² in terms of its prime factors, you simply need to add the x² term, giving you 2 * 2 * 3 * x * x or option C.

A common factor is a number or expression that divides two or more other numbers or expressions without leaving a remainder.

Since 3 is a prime number So the prime factorization of 12 is 2 * 2 * 3.

2. To find the greatest common factor of 8x³ - 12x² + 16x, you can start by factoring out any common factors that the terms share. In this case, all of the terms have a factor of 4x, so you can factor that out to get 4x(2x² - 3x + 4). Therefore, the greatest common factor is 4x or option B.

3. To rewrite 8x³ - 12x² + 16x, pulling out the greatest common factor, you can use the same method as in question 2. The greatest common factor of the terms is 4x, so you can factor that out to get 4x(2x² - 3x + 4) or option A.

4. To find the greatest common factor of 6x²+ 16x + 10, you can factor the expression using the quadratic formula or by factoring the expression as (3x + 5)(2x + 2). Since there is no other common factor between the terms, the greatest common factor is 1 or option A.

5. To factor x² - 4x - 32, you need to find two numbers that multiply to -32 and add to -4. These numbers are -8 and 4. Therefore, x² - 4x - 32 can be factored as (x - 8)(x + 4) or option C.

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The values of the function from the graph are h(7) = 10, h(0) = 9, h(t) = 8, t = 5 and  h(t) = 0, t = 4

Given that the graph is the graph of height as a function of time

To calculate the values of h(t) from the graph of g(x), we need to follow these steps

Using the above, we have the following

h(7) = 10

h(0) = 9

h(t) = 8, t = 5

h(t) = 0, t = 4

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To start, we need to recognize that absolute value is a way of expressing the magnitude of a number, regardless of its sign. Thus, to get rid of the absolute value, we need to consider the two cases in which the expression can have either a positive or a negative sign.

For the case of a positive sign, we can simply write the expression as: 14-(x-6)

For the case of a negative sign, we can multiply both sides of the equation by -1 to get: -14 + (x-6)

Overall, we can express the expression without the absolute value symbol as:

14-(x-6) or -14 + (x-6)

Answer:

The answer is 1.5$

Step-by-step explanation:

Let the price of 1 notebook be x$ and 1 elastic book cover be y$

In first case,

2x+5y=$6.75

2x = $6.75-5y

x=($6.75-5y)/2------------- eqn i

In second case,

4x+2y=$7.50

4×($6.75-5y)/2 +2y=$7.50 [From eqn i]

($27-20y)/2 +2y=$7.50

($27-20y+4y)/2=$7.50

($27-16y)/2=$7.50

$27-16y=$7.50×2

$27-16y=$15

$27-$15=16y

$12=16y

y=$12/16

y=$0.75

The price of single elastic book cover is $0.75

Substituting the value of y in eqn i we get

x=($6.75-5y)/2

x=($6.75-5×$0.75)/2

x=($6.75-$3.75)/2

x=$3/2

x=$1.5

Hence, the price of single notebook is $1.5 Ans

Hey mate, please mark me as brainliest if you got the answer.

Answer:

48 square feet

Step-by-step explanation:

Area of triangle = (1/2) · b · h

(1/2) · 3 · 4 = 6 square feet

There are 2 triangles, so we time 2

6 x 2 = 12 square feet

Area of rectangle = L × W

9 x 4 = 36 square feet

Now we take

12 + 36 = 48 square feet

Answer:

Step-by-step explanation:

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Yes, 8 divides n^2 - 1 when n = 4k + 3.

We are given that n = 4k + 3, where k is an integer. We want to determine if 8 divides n^2 - 1, which means we want to know if there exists an integer m such that n^2 - 1 = 8m.

To start, we can rewrite n^2 - 1 as (n + 1)(n - 1). This is a common trick in number theory problems involving divisibility, and it can often help us factorize expressions in a useful way.

Substituting n = 4k + 3, we get:

(n + 1)(n - 1) = (4k + 3 + 1)(4k + 3 - 1) = (4k + 4)(4k + 2)

Now, we want to determine whether 8 divides this expression. In other words, we want to know if there exists an integer m such that:

(4k + 4)(4k + 2) = 8m

We can simplify the left-hand side of this equation by factoring out 4

4(k + 1)(2k + 1) = 8m

Now, we see that 8 divides the left-hand side of the equation, since it is equal to 4 times an even number. Moreover, (k + 1) and (2k + 1) are both integers, so we can conclude that 8 divides n^2 - 1 when n = 4k + 3.

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Answer:

225 calories

Step-by-step explanation:

1 recipe makes

12 standard sized pancakes

2 pancakes are 1/6 of the recipe

1/6 of the recipe has 150 calories

6 × 150 calories = 900 calories

The full recipe of 12 pancakes has 900 calories

1/2 recipe was made

1/2 recipe has 1/2 × 900 calories = 450 calories

1/2 recipe made 8 pancakes

4 pancakes are half of the half recipe or 1/4 recipe

1/4 × 900 calories = 225 calories

Answer:

Step-by-step explanation:

We can start by assuming that the relationship between the ticket price and attendance is linear, so we can write the equation for the line that connects the two data points we have:

Point 1: (9, 23000)

Point 2: (7, 30000)

The slope of the line can be calculated as:

slope = (y2 - y1) / (x2 - x1)

slope = (30000 - 23000) / (7 - 9)

slope = 3500

So the equation for the line is:

y - y1 = m(x - x1)

y - 23000 = 3500(x - 9)

y = 3500x - 28700

Now we can use this equation to find the attendance for any ticket price. To maximize revenue, we need to find the ticket price that generates the highest revenue. Revenue is simply the product of attendance and ticket price:

R = P*A

R = P(3500P - 28700)

R = 3500P^2 - 28700P

To find the ticket price that maximizes revenue, we need to take the derivative of the revenue equation and set it equal to zero:

dR/dP = 7000P - 28700 = 0

7000P = 28700

P = 4.10

So the ticket price that would maximize revenue is $4.10. However, we need to make sure that this price is within a reasonable range, so we should check that the attendance at this price is between 23,000 and 30,000:

A = 3500(4.10) - 28700

A = 5730

Since 23,000 < 5,730 < 30,000, we can conclude that the ticket price that would maximize revenue is $4.10.

The total interest to be paid over the 5-year period is $403.22.

In a loan with a declining balance, interest is added to the outstanding balance at the start of each period, such as each month or each quarter. The outstanding balance, which decreases as the borrower makes repayments, is used to determine the interest rate. As a result, the borrower gradually pays less interest and more of each payment is used to lowering the main balance due. The lowering balance approach is frequently applied to mortgages, personal loans, and auto loans.

Given that, the quarterly installments are $1899.75.

For the entire year for a total of 5 years the total amount is:

Total amount = 4 * 5 * $1899.75 = $37995

For reducing balance loan the interest is calculated as:

For the first quarter, the interest to be paid is:

interest = (total amount - 0) * (0.14 / 4)

interest = (37995 - 0) * (0.14 / 4) = $94.50

Similarly for the remaining quarters we have:

Second quarter = $87.95

Third quarter: $81.05

Fourth quarter: $73.76

Fifth quarter: $65.96

The total interest to be paid over the 5-year period is:

$94.50 + $87.95 + $81.05 + $73.76 + $65.96 = $403.22

Hence, the total interest to be paid over the 5-year period is $403.22.

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Step-by-step explanation:

For RIGHT triangles, using S-O-H-C-A-H-T-O-A

A sinA= cosB         yes   12/13 = 12/13

B. sinA=tanB          no      12 /13   not equal to  5/12

C. sinB= tanA         no       5/13   not equal to  12/5

D. sinB= cosB         no        5/13   not equal to 12/13

A) sinA= cοsB statement is true. Fοr RIGHT triangles, using S-O-H-C-A-H-T-O-A.

A triangle is a twο-dimensiοnal geοmetric shape that cοnsists οf three straight line segments, called sides, that cοnnect three nοn-cοllinear pοints, called vertices.

Based οn the given image, the fοllοwing statements are true:

1. The triangle is a right triangle, which means that οne οf the angles is a right angle (90 degrees).

2. The side οppοsite the right angle is called the hypοtenuse, which is the lοngest side οf the triangle. In this case, it is the side that is labeled "c".

3. The οther twο sides οf the triangle are called legs. In this case, they are labelled "a" and "b".

4. The Pythagοrean theοrem applies tο this triangle, which states that the sum οf the squares οf the lengths οf the legs is equal tο the square οf the length οf the hypοtenuse. In οther wοrds,  [tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex].

Fοr RIGHT triangles, using S-O-H-C-A-H-T-O-A

A sinA= cοsB         yes     12/13 = 12/13

B. sinA=tanB          nο       12 /13   nοt equal tο  5/12

C. sinB= tanA         nο       5/13   nοt equal tο  12/5

D. sinB= cοsB         nο       5/13   nοt equal tο 12/13.

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Complete Question:

In  Δ ABC , angle C is right Angle

Which Statement must be True?

A sinA= cosB

B. sinA=tanB

C. sinB= tanA

D. sinB= cosB

In the equatiοn , the value οf a and b is 2 and 4.

A mathematical assertiοn that shοws the equality οf twο mathematical expressiοns is what an equatiοn in algebra is defined as. Fοr example, the equatiοn 3x + 5 = 14 is cοmpοsed οf the twο equatiοns 3x + 5 and 14, which are separated by the equal sign.

The equatiοn οf a parabοla in standard fοrm is

y = a(x - h)² + k

where (h, k) are the cοοrdinates οf the vertex and a is the cοefficient οf the x² term.

Here (h, k ) = (- 1, 3 ) and cοefficient οf x² term = 1 , then

=> y = (x - (- 1))² + 3 ,

=> y = (x + 1)² + 3

=> y = x² + 2x + 1 + 3

=> y = x² + 2x + 4

Nοw cοmpare tο y = x² + ax + b

Consequently, a = 2 and b = 4.

Thus, a and b have values of 2 and 4, respectively.

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Two vectors v₁ and v₂ whose sum are (0, -1, -3), where v₁ is parallel to

(5, 0, -4) while v₂ is perpendicular to (5, 0, -4).

v1 = [40/13, -8/13 ]

v2 = [-1/13,  -5/13]

Parallel vectors will be multiples of each other.

Perpendicular vectors will have a dot product of 0.

Let v₁ = [a, b] and v₂ = [c, d]

v₁ || [-5,1]   means  v₁ = N [-5,1] = [a, b]

So  a = -5N  and b = N  

v₂ • [-5,1] = 0 (because they are perpendicular)

So [c, d] • [-5,1] = -5c + d = 0  or  d = 5c    

⇒ v₁ + v₂ = [a, b] + [c, d] = [3, -1]

So  a + c = 3      and     b + d = -1    

Now,

Substitute -5N for

a.  Substitute N for b.    

and Substitute 5c for d.  

This gives us a system of equations with two equations and 2 unknowns:

-5N + c = 3

 N + 5c = -1

Multiply everything in the  bottom equation by 5

-5N + c = 3

5N + 25c = -5

Now add vertically, eliminating the variable N and finding

26c = -2

or, c = -1/13

Now we work our way backwards.

-5N + c = -5N + (-1/13) = 3

-5N = 40/13

(-1/5) (-5N) = (40/13)(-1/5)

and

N = -8/13  and b = N so b = -8/13

d = 5c so d = -5/13

a = -5N, so a = 40/13

So the two vectors are:

v₁ = [40/13, -8/13 ]

v₂ = [-1/13,  -5/13]

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The frequency in hertz of electromagnetic radiation which has a wavelength of 585nm is 5.13×10¹⁴ Hz.

The frequency(f) of electromagnetic radiation is related to its wavelength (λ) by the equation : f = c/λ;

Where, c = the speed of light in a vacuum, which is  3.00×10⁸ m/s.

So, to calculate the frequency of radiation with a wavelength of 585.0 nm, we need to convert the wavelength to meters:

⇒ 585.0 nm = 585.0×10⁻⁹ m;

Substituting the value in the above formula,

We get,

⇒ f = c/λ = 3.00×10⁸/(585.0×10⁻⁹)

⇒ 5.13×10¹⁴ Hz.

Therefore, the frequency of electromagnetic radiation is approximately 5.13 × 10¹⁴ Hz.

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i) Find the mean, median, and mode of the frequency table as follows:

ii) The average that justifies the teacher's statement congratulating the class that 'over three quarters were above average' is the average mark of 10, which is 5.

The mean refers to the average or the quotient of the total values divided by the number of items.

The median is the middle value in the data, which occurs with marks 8 for the 13th and 14th students.

The mode is the value that occurs most frequently, which is 3 which occurs 6 times.

Frequency Table:

Mark   Frequency  Cumulative Frequency

3              6                            18 (0 + 3 x 6)

4              3                            30 (18 + 4 x 3)

5              1                            35 (30 + 5 x 1)

6              2                           47 (35 + 6 x 2)

7              0                           47 (47 + 7 x 0)

8              5                           87 (47 + 8 x 5)

9              5                         132 (87 + 9 x 5)

10            4                         172 (132 + 10 x 4)

Mean = 6.6 (172/26)

Median = 8

Mode = 3

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The correct options are:

The data contain a sample of multiple individuals.

The data are collected  are approximately at  the same point in time.

Cross-sectional data sets are a type of research design used in statistical analysis. They consist of a sample of multiple individuals or entities observed at a single point in time. One of the characteristics of cross-sectional data sets is that they do not involve any observation or measurement over time, making them different from longitudinal or time-series data sets.

One advantage of cross-sectional data sets is that they are relatively easy and inexpensive to collect. It can be assumed that the data were obtained through a random sampling of the underlying population, making them representative of the larger population. However, these data require attention to the frequency at which they are collected (weekly, monthly, yearly, etc.), as the timing of data collection can impact the results.

Overall, cross-sectional data sets can provide a snapshot of a population or phenomenon at a specific point in time, making them useful for a wide range of research questions in social, economic, and political sciences.

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Answer:

(a) 7.97E33 = 7.97 x 10^33

(b) 7.6E-37 = 7.6 x 10^-37

Step-by-step explanation:

In scientific notation, a number is written as the product of a decimal number between 1 and 10 and a power of 10. The decimal number is called the mantissa, and the power of 10 is called the exponent. The mantissa indicates the significant digits of the number, while the exponent tells us the size of the number relative to 1. In the given calculator computations, the numbers are already written in scientific notation, where "E" represents the exponent notation. We just need to convert them into the standard form of scientific notation by expanding the notation as a product of the mantissa and the power of 10.

Answer:

Step-by-step explanation:

1)  b      (acute is less than 90)

2)  a     (obtuse: more than 90, less than 180)

3)  c

4) c

Answer:

1. NOM, JOK, KOL

2. MOL, NOK, MOJ

3. NOJ, JOL

4. NOL, MOK

Answer: Yes;

Step-by-step explanation: The function is linear in that the X values are increasing by 2 and the Y values are increasing by 3.  

Answer:

Not linear

Step-by-step explanation:

X | -7 | -5 | -3 | -1 | 0

Y | 11 | 14 | 17 | 20 | 23

-5 - (-7) = +2

14 - 11 = +3

-3 - (-5) = +2

17 - 14 = +3

-1 - (-3) = +2

20 - 17 = +3

0 - (-1) = +1

23 - 20 = +3

There is a linear function graphed that would pass through the first 4 points, but not the last one;

You can discern this as there is a common pattern we can identify, everytime the x-value increases by 2 the y-value increases by 3, except with the last coordinates;

The rate of change, also known as the gradient, is the change in x divided by the change in y (Δx/Δy);

In the case of the first 4 points, the rate of change would be 3/2 or 1.5;

This simply means when x increases by 1, y increases by 1.5, IF the function is linear;

However, as mentioned the last point doesn't fit the pattern;

There, we can see the y-value increases by 3 when the x-value only increases by 1;

This means the point (0, 23) isnt on the graph of the function, the points (0, 21.5) and (1, 23), on the other hand, would be.

Answer: 200 texts

Step-by-step explanation:

          If they paid the same amount, we can set the equations equal to each other and solve for x using inverse operations.

    Given:

0.1x + 10 = 0.15x

    Subtract 10 and 0.15x from both sides of the equation:

-0.05x = -10

    Divide both sides of the equation by -0.05:

x = 200

The probability of getting a measurement above -1.12°C is 0.8686, or almost 87%.

Typically, the probability is defined as the ratio of favorable events to all other outcomes in the sample space.Probability of an event P(E) = (Number of favorable outcomes) is how it is stated (Sample space).

We must determine the area under the normal distribution curve to the right of -1.12°C in order to determine the likelihood of getting a reading higher than that value.

We may convert the value of -1.12°C to a z-score, which represents the amount of standard deviations from the mean, using a calculator or a typical normal distribution table:

z = (x - μ) / σ

z = (-1.12 - 0) / 1.00

z = -1.12

The area under the normal distribution curve to the right of a particular z-score can also be found using a calculator. For instance, we can type the following into a calculator that has a normal distribution function:

P(Z > -1.12)

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This function gives the same result as the previous function we derived, but it expresses the calculation as a composition of the two separate functions that represent each raise separately.

A function is a set of instructions or statements that perform a specific task or computation and returns a value or result. In programming, a function is typically defined using a specific syntax and can be called or invoked by other parts of the program to carry out a particular operation.

Functions are used in programming to break down complex tasks into smaller, more manageable pieces of code, which can then be reused in multiple parts of the program. They can also help to improve the organization, readability, and maintainability of code.

by the question.

let the worker's original hourly wage be represented by the variable x. Then the function for the worker's new hourly wage when the 20% raise is applied before the $2 raise can be expressed as follows:

[tex]f(x) = (1.20x) + 2[/tex]

In this function, the expression (1.20x) represents the worker's hourly wage after the 20% raise is applied, and the additional $2 raise is added to this amount to determine the final hourly wage.

We can also express this function using function composition notation. Let g(x) represent the function that applies the 20% raise to the worker's hourly wage and let h(x) represent the function that adds the $2 raise to the resulting hourly wage. Then we have:

[tex]g(x) = 1.20x[/tex]

[tex]h(x) = x + 2[/tex]

Using function composition, we can write the function for the worker's new hourly wage as:

[tex]f(x) = h(g(x)) = h(1.20x) = 1.20x + 2[/tex]

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Answer:

A. Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer