The product of five rational numbers is positive. At most, of these rational numbers can be negative

A.1
B.2
C.4
D.5​

The standard form is 6x⁴+5x³-2x-15.

Here the term polynomial is characterized as an expression that is composed of variables, exponents, and constants, that are combined utilizing numerical operations such as subtraction,  addition, division, and multiplication. the standard form of a polynomial is :

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ +aₙ₋₂xⁿ⁻² + ………………. + a₁x + a₀  

Where aₙ, aₙ₋₁, aₙ₋₂, ……………………, a₁, a₀  are the coefficients of xⁿ, xⁿ⁻¹, xⁿ⁻², ….., x and are to a real number.

The polynomial given to us is (3x² + x − 5) (2x² + x + 3)

=3x²(2x² + x + 3) + x (2x² + x + 3) − 5 (2x² + x + 3)

= 6x⁴ + 3x³+9x²+2x³+x²+3x-10x²-5x-15

= 6x⁴+3x³+2x³+9x²+x²-10x²+3x-5x-15

= 6x⁴+5x³-2x-15

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Answer:

x = 15

z = 25

Step-by-step explanation:

In \( \odot\: O, \: \angle MLK \) is inscribed in a semicircle.

\( \therefore m\angle MLK =90\degree \)

\( \therefore 6x\degree =90\degree \)

\( \therefore 6x =90\)

\( \therefore x =\frac{90}{6}\)

\( \therefore x =15\)

In \( \odot\: A, \: \angle PRQ\) is inscribed in a semicircle.

\( \therefore m\angle PRQ =90\degree \)

\( \therefore (4z-10)\degree =90\degree \)

\( \therefore 4z-10 =90\)

\( \therefore 4z =100\)

\( \therefore z =\frac{100}{4}\)

\( \therefore z =25\)

The graph of the function f(x) = 5(2)^x is an upward-sloping exponential curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing the x-axis.

The function f(x) = 5(2)^x represents exponential growth. Let's analyze its graph.

As x increases, the value of 2^x grows exponentially. Multiplying it by 5 further amplifies the growth. Here are a few key points to consider:

When x = 0, 2^0 = 1, so f(0) = 5(1) = 5. This is the y-intercept of the graph, meaning the function passes through the point (0, 5).

As x increases, 2^x grows rapidly. For positive values of x, the function will increase quickly. As x approaches positive infinity, 2^x grows without bound, resulting in the function also growing without bound.

For negative values of x, 2^x approaches zero. However, the function is multiplied by 5, so it will not reach zero. Instead, it will approach y = 0, but the graph will never touch or cross the x-axis.

The function is always positive since 2^x is positive for any value of x, and multiplying by 5 does not change the sign.

Based on these observations, we can conclude that the graph of f(x) = 5(2)^x will be an exponential growth curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing or touching the x-axis.

The graph will have a smooth curve that rises steeply as x increases. The rate of growth will be determined by the base, in this case, 2. The larger the base, the steeper the curve. The function will approach but never reach the x-axis as x approaches negative infinity.

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The value of the probability is 0.0714.

According to the statement

We have to find that the students' favorite color was red.

So, For this purpose, we know that the

Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions.

From the given information:

A bag 2 contains 2 red marbles, 2 green marbles, and 4 blue marbles.If we choose a marble, then another marble without putting the first one back in the bag.

Then

A bag contains 2+2+4=8 marbles in total.

We have to search out the probability of the subsequent events:

1. we've to settle on a red marble from the bag that contains 8 marbles in total, 2 of them are red

2. we've to decide on a green marble from the bag that contains 7 marbles in total, 2 of them are green.

The probability of the primary event to happens is:

P(one) = 2/8 = 1/4.

The probability of the second event to happen is:

P(two) = 2/7

The probability of both events to happen is:

P = P(one) * P(two)

P = 1/4 * 2/7

P = 1/14

P = 0.0714.

So, The value of the probability is 0.0714.

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Based on the information, the hedgehogs have made the better offer

The Turkeys are offering a yearly salary of S(t) = 90000t over the course of eight years. To calculate the present value of each payment, an equation is used:

PV = S(t) * e^(-rt)

The variable "r" denotes a continuous interest rate of 0.03. "t" equals the years from t=0 to the time of payment. By this method, the current value of the Turkeys' offer can be assessed as follows:

PV(Turkeys) = ∫[0,8] 90000t * e^(-0.03t) dt

= 736,918.63

Comparatively, the Hedgehogs propose a yearly salary of S(t) = 80000t for nine years. Utilizing the formula employed in the previous scenario, the following result is derived:

PV(Hedgehogs) = ∫[0,9] 80000t * e^(-0.03t) dt

= 753,472.49

Given these calculations and the presumption that Rogelio may earn a continuous interest rate of 3% on his funds, it would best serve him to accept the Hedgehogs’ item offer since they have offered him a more favorable contract based on the accumulated present value of both contracts.

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The slope of the line passing through the points (-6, 1) and (8, -7) is -4/7.

Slope is simply expressed as change in y over the change in x.

Slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given the data in the question;

Point A: ( -6,1 )

x₁ = -6

y₁ = 1

Point B: ( 8,-7 )

x₂ = 8

y₂ = -7

Now, plug the given x and y values into the slope formula above and simplify:

\(Slope\ m = \frac{y_2 - y_1}{x_2 - x_1} \\\\Slope\ m = \frac{-7 - 1}{8 - (-6)} \\\\Slope\ m = \frac{-7 - 1}{8 + 6} \\\\Slope\ m = \frac{-8}{14} \\\\Slope\ m = -\frac{4}{7}\)

Therefore, the slope of the line is -4/7.

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The sample standard deviation for the given data set is approximately 2.77.

Let's calculate the sample standard deviation for the given data set:

Calculate the mean

Sum of all data points = 8 + 5 + 2 + 4 + 4 + 9 + 9 + 7 + 3 + 5 + 2 + 4 + 4 + 7 + 10 + 5 + 1 + 8 + 9 + 8 + 3 + 3 + 8 + 7 = 137

Mean = Sum of all data points / Number of data points = 137 / 24 = 5.71 (rounded to two decimal places)

Subtract the mean from each data point and square the result

(8 - 5.71)^2 = 2.4361

(5 - 5.71)^2 = 0.5041

(2 - 5.71)^2 = 15.1361

(4 - 5.71)^2 = 2.9241

(4 - 5.71)^2 = 2.9241

(9 - 5.71)^2 = 10.3961

(9 - 5.71)^2 = 10.3961

(7 - 5.71)^2 = 1.6641

(3 - 5.71)^2 = 8.2561

(5 - 5.71)^2 = 0.5041

(2 - 5.71)^2 = 15.1361

(4 - 5.71)^2 = 2.9241

(4 - 5.71)^2 = 2.9241

(7 - 5.71)^2 = 1.6641

(10 - 5.71)^2 = 23.7761

(5 - 5.71)^2 = 0.5041

(1 - 5.71)^2 = 21.8441

(8 - 5.71)^2 = 2.4361

(9 - 5.71)^2 = 10.3961

(8 - 5.71)^2 = 2.4361

(3 - 5.71)^2 = 8.2561

(3 - 5.71)^2 = 8.2561

(8 - 5.71)^2 = 2.4361

(7 - 5.71)^2 = 1.6641

Find the sum of all the squared differences

Sum of squared differences = 2.4361 + 0.5041 + 15.1361 + 2.9241 + 2.9241 + 10.3961 + 10.3961 + 1.6641 + 8.2561 + 0.5041 + 15.1361 + 2.9241 + 2.9241 + 1.6641 + 23.7761 + 0.5041 + 21.8441 + 2.4361 + 10.3961 + 2.4361 + 8.2561 + 8.2561 + 2.4361 + 1.6641 = 174.07 (rounded to two decimal places)

Divide the sum by the number of data points minus 1

Variance = Sum of squared differences / (Number of data points - 1) = 174.07 / (24 - 1) = 7.67 (rounded to two decimal places)

Take the square root of the variance

Sample Standard Deviation = √(Variance) = √(7.67) = 2.77 (rounded to two decimal places)

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The two lines are perpendicular because the product of their slopes is equal to 1

The slope-intercept form of a line is represented by y=mx+c where m=slope and c is the y-intercept of the line

Given here: The two lines  x-3y =15 and y = -3y + 4

Rewriting the given expressions in slop-intercept form we get

y=x/3-5

y = -3y + 4

The respective slopes are given as 1/3 and -3

Thus the product of the two slopes is 1/3×-3=-1

Hence, The given lines are perpendicular.

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h = hours till they're 58 miles apart

Check the picture below.

so they're travelling in opposite directions, however the combined distances is 58 miles at say "h" hours, by the time that happend Hopi has been walking "h" hours and Brittany has also being walking "h" hours too.

Since the combined distance is 58 miles than if Hopi has covered "m" miles then Brittany covered the slack of "58 - m".

\({\Large \begin{array}{llll} \underset{distance}{d}=\underset{rate}{r} \stackrel{time}{t} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lcccl} &\stackrel{miles}{distance}&\stackrel{mph}{rate}&\stackrel{hours}{time}\\ \cline{2-4}&\\ Hopi&m&15&h\\ Brittany&58-m&14&h \end{array}\hspace{5em} \begin{cases} m=(15)(h)\\\\ 58-m=(14)(h) \end{cases} \\\\[-0.35em] ~\dotfill\)

\(\stackrel{\textit{substituting on the 2nd equation}}{58-(\stackrel{m}{15h})=14h}\implies 58=29h\implies \cfrac{58}{29}=h\implies \boxed{2=h}\)

Answer:

I hope it helped you

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Answer:

see explanation

Step-by-step explanation:

a

7² = 7 × 7 = 49

b

6³ = 6 × 6 × 6 = 36 × 6 = 216

c

12² = 12 × 12 = 144

d

\(2^{4}\) = 2 × 2 × 2 × 2 = 4 × 4 = 16

e

10³ = 10 × 10 × 10 = 100 × 10 = 1000

36 - k = 4 + k

36 - 4 = k + k

32 = 2k

32/2 = k

16 = k

Steps:

Step 1: Simplify both sides of the equation

36−k=4+k

36+−k=4+k

−k+36=k+4

Step 2: Subtract k from both sides

−k+36−k=k+4−k

−2k+36=4

Step 3: Subtract 36 from both sides

−2k+36−36=4−36

−2k=−32

Step 4: Divide both sides by -2

−2k/−2 = −32/−2

Description:

Since we are trying to find the value of K, we need to simplify both sides of the equation. After that your answer will come as −k+36=k+4.  Now the second step is to  subtract k from both sides, your equation will come as −2k+36=4.  Thirdly we need to subtract 36 from both sides, your equation will come as −2k=−32.  Now you need to divide both sides by -2. After you do that, you will get your answer which is k=16.

Answer: k=16

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Hope this helps.

Answer: Therefore, the set of points that have the smallest SSE is (1, 5) and (7, 3), and the line of fit would go through these two points to best fit the data.

Step-by-step explanation:

To determine the line of best fit, we need to find the equation of the line that passes through the given points. This can be done using the slope-intercept form of the equation of a line, y = mx + b, where m is the slope of the line and b is the y-intercept.

For each set of points, we can find the slope using the formula:

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

and then use one of the points to solve for b.

(6, 4) and (9, 1):

m = (1 - 4) / (9 - 6) = -3/3 = -1

Using point (6, 4):

4 = -1(6) + b

b = 10

So the equation of the line is y = -x + 10.

(3, 5) and (10, 1):

m = (1 - 5) / (10 - 3) = -4/7

Using point (3, 5):

5 = (-4/7)(3) + b

b = 41/7

So the equation of the line is y = (-4/7)x + 41/7.

(1, 8) and (10, 1):

m = (1 - 8) / (10 - 1) = -7/9

Using point (1, 8):

8 = (-7/9)(1) + b

b = 71/9

So the equation of the line is y = (-7/9)x + 71/9.

(1, 5) and (7, 3):

m = (3 - 5) / (7 - 1) = -1/3

Using point (1, 5):

5 = (-1/3)(1) + b

b = 16/3

So the equation of the line is y = (-1/3)x + 16/3.

To determine which two points would a line of fit go through to best fit the data, we need to choose the set of points that have the smallest average distance between the points and the line. One way to measure this is by calculating the sum of the squared errors (SSE) between the points and the line for each set of points, and choosing the set with the smallest SSE.

Using the equations we found for each set of points, we can calculate the SSE:

(6, 4) and (9, 1):

SSE = (4 - (-1(6) + 10))^2 + (1 - (-1(9) + 10))^2 = 29

(3, 5) and (10, 1):

SSE = (5 - (-4/7)(3) + 41/7)^2 + (1 - (-4/7)(10) + 41/7)^2 = 16.9

(1, 8) and (10, 1):

SSE = (8 - (-7/9)(1) + 71/9)^2 + (1 - (-7/9)(10) + 71/9)^2 = 28.7

(1, 5) and (7, 3):

SSE = (5 - (-1/3)(1) + 16/3)^2 + (3 - (-1/3)(7) + 16/3)^2 = 7.6

Answer:

The inverse is,

\(f^{-1}(x) = \sqrt[5]{x} - 3\)

Step-by-step explanation:

f(x) = (x+3)^5

Finding the inverse,

\(f(x) = (x+3)^5\\or,\\y = (x+3)^5\)

We replace x with y and vice versa

so,

\(x = (y+3)^5\)

Solving for y,

the the 5th root,

\(\sqrt[5]{x} = y + 3\\y = \sqrt[5]{x} - 3\)

hence the inverse function is,

\(f^{-1}(x) = \sqrt[5]{x} - 3\)

Step-by-step explanation:

f(x)=(x+3)×5

Y=5X+15

now

interxchanging x and y we get,

x=5y+15

5y=x-15

y=x-15/5

therefor f~1(x)=x-15/5

Answer:

Length of rectangular strip = 12

area of rectangular strip =   2*12 = 24

Area of square =  x^2 = 12^2 = 144

Step-by-step explanation:

Area of square x^2

area of rectangle is given by length * width

Length of rectangular strip = x

width of rectangular strip = 2

area of rectangular strip =  length * width = 2*x = 2x

Area of square piece of paper when  rectangular strip is taken away from it

= Area of square - area of rectangular strip

= \(x^2 -2x\)

It is given that Area of square piece of paper when  rectangular strip is taken away from it is 120 square units.

Thus,

\(x^2 -2x = 120\\=> x^2 -2x -120 = 0\\=> x^2 -12x +10x-120\\=> x(x-12) +10(x-12)\\=> (x+10)(x-12) = 0\\\\\)

Thus,

either x+10 = 0 or x -12= 0

x = -10 or x = 12

but length cannot  be negative hence neglecting x = -10

hence value of x is 12.

Hence,

Length of rectangular strip = 12

area of rectangular strip =   2*12 = 24

Area of square =  x^2 = 12^2 = 144

Volume of space inside shop is 288 m³

Given that;

Dimensions of rectangle = 8 meter , 12 meter , 3 meter

Find:

Volume of space inside shop

Computation:

Volume of rectangle = [l][b][h]

Volume of rectangle = Volume of space inside shop

So,

Volume of space inside shop = [8][12][3]

Volume of space inside shop = [96][3]

Volume of space inside shop = 288 m³

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Answer:

1/12 or 0.0833

Step-by-step explanation:

1/2 divided by 6

1/2x6

1/12!

Answer: Fraction Form: 1/12

              Decimal Form: 0.0833

              Expanded Decimal Form: 0.08333333333

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Step-by-step explanation:

the nth root can be written as exponent of 1/n.

so, we have here

\( {( {n}^{4} + 3) }^{ \frac{1}{5} } - 12\)

Answer:

growth is usually reffered to as physical growth happening in size while development happens more gradually and happens mentally.

Step-by-step explanation:

idk if u meant pyscologically or not but that is my understanding.

Answer:

That is a rectangle.

Step-by-step explanation:

We cannot say a circle because it is the shape that is being cross sectioned.

Answer:

f(x) = x³ - 4x + 6

f'(x) = 3x² - 4

a) f'(2) = 3(2²) - 4 = 12 - 4 = 8

6 = 8(2) + b

6 = 16 + b

b = -10

y = 8x - 10

b) 3x² - 4 = 0

3x² = 4, so x = ±2/√3 = ±(2/3)√3

= ±1.1547

f(-(2/3)√3) = 9.0792

f((2/3)√3) = 2.9208

c) 3x² - 4 = 8

3x² = 12

x² = 4, so x = ±2

f(-2) = (-2)³ - 4(-2) + 6 = -8 + 8 + 6 = 6

6 = -2(8) + b

6 = -16 + b

b = 22

y = 8x + 22

f(2) = 6

y = 8x - 10

The equation perpendicular to the tangent is y = -1/8x + 25/4

-The smallest slope on the curve is 2.92

The curve has the smallest slope at the point (1.15, 2.92)

The equations at tangent points are y = 8x + 16 and  y = 8x - 16

From the question, we have the following parameters that can be used in our computation:

y = x³ - 4x + 6

Differentiate

So, we have

f'(x) = 3x² - 4

The point is (2, 6)

So, we have

f'(2) = 3(2)² - 4

f'(2) = 8

The slope of the perpendicular line is

Slope = -1/8

So, we have

y = -1/8(x - 2) + 6

y = -1/8x + 25/4

We have

f'(x) = 3x² - 4

Set to 0

3x² - 4 = 0

Solve for x

x = √[4/3]

x = 1.15

So, we have

Smallest slope = (√[4/3])³ - 4(√[4/3]) + 6

Smallest slope = 2.92

So, the smallest slope is 2.92 at (1.15, 2.92)

Here, we set f'(x) to 8

3x² - 4 = 8

Solve for x

x = ±2

Calculate y at x = ±2

y = (-2)³ - 4(-2) + 6 = 6: (-2, 0)

y = (2)³ - 4(2) + 6 = 6: (2, 0)

The equations at these points are

y = 8x + 16

y = 8x - 16

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Answer:

Step-by-step explanation:

The best estimate for the volume of a coffee mug would be the volume of a cylinder. The volume of a cylinder is:

\(V=\pi r^2h\)

The volume can be thought of as the area of a circle, being scaled by the height. The area of the circular base of the mug, multiplied by the height of the mug.

Answer:

To calculate Karl Pearson's coefficient of skewness, we need to use the formula:

Skewness = 3 * (Mean - Mode) / Standard Deviation

Given the Mean = 73.19, Mode = 79.56, and Variance = 16, we need to find the Standard Deviation first.

Standard Deviation = √Variance = √16 = 4

Now we can substitute the values into the formula:

Skewness = 3 * (73.19 - 79.56) / 4

Skewness = -6.37 / 4

Skewness = -1.5925

Rounded to four decimal places, the Karl Pearson's coefficient of skewness for the given values is approximately -1.5925.

True, While "-1000 2/3" is not a real fraction, it can be represented as the improper fraction -2998/3.

The statement "-1000 2/3 is not a real fraction" is true. A real fraction is a mathematical expression that represents a ratio of two real numbers. In a fraction, the numerator and denominator are both real numbers, and they can be positive, negative, or zero.

In the given statement, "-1000 2/3" is not a valid representation of a fraction. The presence of a space between "-1000" and "2/3" suggests that they are separate entities rather than being part of a single fraction.

To represent a mixed number (a whole number combined with a fraction), a space or a plus sign is typically used between the whole number and the fraction. For example, a valid representation of a mixed number would be "-1000 2/3" or "-1000 + 2/3". However, without the proper formatting, "-1000 2/3" is not considered a real fraction.

It's important to note that "-1000 2/3" can still be expressed as an improper fraction. To convert it into an improper fraction, we multiply the whole number (-1000) by the denominator of the fraction (3) and add the numerator (2). The result would be (-1000 * 3 + 2) / 3 = (-3000 + 2) / 3 = -2998/3.

In conclusion, while "-1000 2/3" is not a real fraction, it can be represented as the improper fraction -2998/3.

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The motion of the skier from point A to point B, then to point C, and then point D, indicates that the distances and displacement traveled are;

a) The distance from A to C is 280 meters

b) The distance from point A to point D is 360 meters

c) The displacement of the skier from point B to point C is 120 meters.

Displacement is the change in the position or location of the object.

The possible diagram of the skier in the question can be presented as follows;

A \({}\)                            C                                                   D                          B

t = 0 min      \({}\)         t = 2 min                                       t = 3 min            \({}\)t = 1 min

ʂ         \({}\)                     ʂ                                                     ʂ                           ʂ

-|-------|------|------|------|-----|------|-----|------|------|-------|-------|--------|-------|-------|-----

0           \({}\)   20           40         60          80            100    120    140           160

The above diagram indicates that we get;

a) The distance from point A (t = 0 min) to point C(t = 2 min) is can be found as follows;

The skier moves from point A to point B then from point B to point C, therefore,

The distance from point A to point B = 160 m - 0 m = 160 m

The distance from point B to point C = 160 m - 40 m = 120 m

The distance from point A to point C is therefore;

b) The distance from point A to point D = The distance from point A to point C + The distance from point C to point D as follows;

AD = AC + CD

CD = 120 m - 40 m = 80 m

Therefore;

AD = 280 m + 80 m = 360 m

c) The displacement of the skier from point B (t = 1 min) to C (t = 2 min) can be found as follows;

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Report that other guy smh... but im not quite too sure but i believe the answer you were looking for was C. The total cost will increase by 4$ every 3 games bowled :D

3. The factored polynomial is p(x) = (x - 1)(x - 5)

4. The factored polynomial is p(x) = (x + 3)²

A polynomial is a mathematical expression in which the power of the unknown is greater than or equal to 2.

3. To factorize the polynomial using the graph, we see that the polynomial cuts the x - axis at x = 1 and x = 5.

This implies that its factors are (x - 1) and (x - 5)

So, the factored polynomial is p(x) = (x - 1)(x - 5)

4. To factorize the polynomial using the graph, we see that the polynomial touches the x - axis at only one point x = -3. So,it has repeated roots

This implies that its factors are (x - (-3)) = (x + 3) twice

So, the factored polynomial is p(x) = (x + 3)²

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Answer:

29 degrees

Step-by-step explanation:

the whole Angle equals 73 degrees

73° - 44° = 29°

Answer:

Step-by-step explanation:

9 - a = 9 -a

9 - a + 10 = 19 - a