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The HCF of the polynomials x^6 -  3x^4+ 3x² - 1 and x³+ 3x^2+ 3x+ 1 is (x+1)^3.

Let us suppose the Polynomials to be f(x) = x^6 - 3x^4 + 3x^2 - 1  And polynomial g(x) = x^3 + 3x^2 + 3x + 1

Now,  According to the question,

we have f(x) = x^6 - 3x^4 + 3x^2 - 1  

Adding and subtracting 2^x2,

⇒ x^6 - x^4 - 2x^4 + 2^x2 + x^2 - 1

⇒ x^4(x^2 - 1) - 2x^2(x^2 - 1) +1(x^2 + 1)

⇒ (x^2 - 1)(x^4 - 2x^2 + 1)

⇒ (x + 1)(x - 1)(x^4 - 2x^2 + 1)

⇒ (x + 1)(x - 1)(x2 - 1)^2

⇒ (x + 1) (x - 1) (x + 1)^2 (x - 1)2

⇒ (x + 1)^3(x - 1)^3      ----(1)

And, g(x) = x^3 + 3x^2 + 3x + 1

⇒ x^3 + x^2 + 2x^2 + 2x + x + 1

⇒ x^2(x + 1) + 2x(x + 1) + 1(x + 1) ⇒ (x + 1)(x^2 + 2x + 1)

⇒ (x + 1) (x + 1)2 ⇒ (x + 1)3      ----(2)

Now, The HCF of f(x) and g(x) is the common factor of equations (1) and (2)

⇒ (x + 1)^3 .

∴ The HCF of the polynomials (x^6 - 3x^4 + 3x^2 - 1) and (x^3 + 3x^2 + 3x + 1) is (x + 1)^3.

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Answer:

There is no answer.

Step-by-step explanation:

4(2x+3) = 3+8x-11

Group together common terms on the right & use the distributive property on the left.

4(2x) + 4(3) = (3-11) +8x

8x + 12 = 8x -8

Combine like terms. Subtract 8x from both sides.

12 = -8

A relative frequency distribution shows C. the fraction or percentage of observations in each class interval.

A relative frequency is a proportional value that indicates the relative number of a specific event in a total number of observations.

A relative frequency uses percentages, proportions, and fractions to show the value of the chosen or expected outcome.

With a relative frequency distribution, it does not merely show the number of observations in each class interval or set of data.

Thus, a relative frequency distribution indicates Option C because it compares one value against other observations in relative terms.

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The general value of theta which satisfies the equation (cosФ+isinФ)(cos2Ф+isin2Ф)..............(cosnФ+isinnФ) is one is 4π/(n+1)

Real and imaginary numbers:

A real number can be a natural number, a whole number, an integer, a rational number, or an irrational number. But an imaginary number is the product of a real number and "i" where i = √(-1). For example, √(-9) = √(-1) . √9 = i (3) = 3i

cosФ+isinФ = [tex]e^{i0}[/tex]

cos2Ф+isin2Ф = [tex]e^{2i0}[/tex]

............................

cosnФ+isinnФ= [tex]e^{in0}[/tex]

Now,

[tex]e^{i0}[/tex]×[tex]e^{2i0}[/tex]×.......[tex]e^{in0}[/tex] =1

[tex]e^{i0(1+2+....n)}[/tex]  = 1

[tex]e^{i0(n*(n+1)/2)}[/tex] =1

So

cos(n*(n+1)/2)Ф+isin(n*(n+1)/2)Ф =1

Now comparing the real and the imaginary part we get that

cos(n*(n+1)/2)Ф = 1

n*(n+1)/2Ф = 2nπ

Ф = 4π/(n+1)

Therefore, the general value of theta which satisfies the equation (cosФ+isinФ)(cos2Ф+isin2Ф)..............(cosnФ+isinnФ) is one is 4π/(n+1).

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The correct question should be:

find the general value of theta which satisfies the equation (cosФ+isinФ)(cos2Ф+isin2Ф)..............(cosnФ+isinnФ) equals to one.

Answer: B )" AAS congruence theorem" an be used to prove that the triangles are congruent .

AAS congruence theorem tells that if two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle then the triangles are congruent.

Therefore option B is the correct answer. "AAS congruence" theorem an be used to prove that the triangles are congruent

Answer:

| AD | ≈ 48 m

Step-by-step explanation:

Answer: 297492

Step-by-step explanation:

[tex]\huge\text{Hey there!}[/tex]

[tex]\mathsf{10 [12^2 + (4 \times 2)] \div 8^2}[/tex]

[tex]\mathsf{= 10[12^2 + (8)] \div 8^2}[/tex]

[tex]\mathsf{= 10[12^2 + 8)] \div 8^2}[/tex]

[tex]\mathsf{= 10[12 \times 12 + 8] \div 8 \times 8}[/tex]

[tex]\mathsf{= 10[144 + 8] \div 64}[/tex]

[tex]\mathsf{= 10[152] \div 64}[/tex]

[tex]\mathsf{= 10(152) \div 64}[/tex]

[tex]\mathsf{= 1,520 \div 64}[/tex]

[tex]\mathsf{= \dfrac{1,520}{64}}[/tex]

[tex]\mathsf{= \dfrac{1,520 \div 16}{64 \div 16}}[/tex]

[tex]\mathsf{= \dfrac{95}{4}}[/tex]

[tex]\mathsf{= 23 \dfrac{3}{4}}[/tex]

[tex]\huge\text{Therefore, your answer is:}[/tex]

[tex]\huge\boxed{\mathsf{= \dfrac{95}{4}\ or \ 23 \dfrac{3}{4}\ or \ even\ 23.75}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

The equation of the quadratic function is (b) C(t) = (t - 4)²

From the question, we have the table of values that can be used in our computation:

A quadratic equation is represented as

f(x) = a(x - h)² + k

Where

Vertex = (h, k)

From the graph, we have the vertex to be

(h, k) = (4, 0)

Substitute (h, k) = (4, 0) in f(x) = a(x - h)² + k

So, we have

f(x) = a(x - 4)²

Also, from the graph, we have the point (2, 4)

This means that

a(2 - 4)² = 4

So, we have

4a = 4

Divide both sides by 4

a = 1

Substitute a = 1 in f(x) = a(x - 4)²

f(x) = (x - 4)²

Rewrite as C(t)

C(t) = (t - 4)²

Hence, the equation is C(t) = (t - 4)²

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its C(t) = (x − 4)2

use desmos graphing its very helpful

Multiplicity of 3 is the number of times that the value 3 appears as a root of an equation  is the multiplicity of 3 is 3

Multiplicity of 3 (also known as a 3-fold root) is a mathematical concept used to describe the number of times a certain value appears as a root (or solution) of an equation. Specifically, it refers to the number of times a certain value appears as a root of a given equation when the equation is solved.

For example, the equation x2 = 9 has three solutions: x = 3, -3, and 0. This means that the multiplicity of 3 is 3, because 3 appears three times as a root of the equation. The general formula used to calculate the multiplicity of a given root is:

Multiplicity of x = Number of times x appears as a root of the equation

In this case, the multiplicity of 3 is 3, because 3 appears as a root of the equation three times. Similarly, if there are two solutions, x = -2 and x = 0, the multiplicity of -2 is 2, because -2 appears as a root of the equation twice.

In summary, multiplicity of 3 is the number of times that the value 3 appears as a root of an equation when the equation is solved. In the example above, the multiplicity of 3 is 3.

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The gradient of y=-4x+9 is -4. The gradient of a line is the measure of its steepness.

It is calculated by taking the derivative of the line. When taking the derivative of a linear equation, the derivative is the coefficient of the variable (x). In this case, the coefficient of x is -4. Therefore, the gradient of y=-4x+9 is -4.

To calculate this gradient, we need to use the power rule of derivatives. This rule states that if y = ax^n, then the derivative of y is nax^(n-1). In this case, a=-4 and n=1. Therefore, the derivative of y=-4x+9 is -4*1^(1-1), which simplifies to -4.

To calculate the gradient of y=-4x+9, we applied the power rule of derivatives. The coefficient of x in this equation was -4, so the derivative of y was -4. Therefore, the gradient of y=-4x+9 is -4.

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Answer:

y = 5x + 10

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

We know slope 5 and y-intercept 10

So, our equation is

y = 5x + 10

The value of "i" squared or 1 i by 1 i results in:

(-1)

Among the numerical sets there is one that we call complex numbers, which include values that are not real, such as "i" a letter that denotes that it is an imaginary number.

Numerical sets are groupings of numerical values that have a particularity in common, they can be integers, decimals, fractions, among others.

In function to this type of numbers there is a particular value and it is the "i" which is the result of the square root of -1, then we have:

√(-1) x √(-1) = i²

[√(-1)]² = i²

(-1) = i²

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A line bisector divides a given line into two equal sections or parts. Thus the required proof to show that BC ≅ DF is stated below:

A given line is said to have been bisected if and only if it is divided into two equal parts by a constructed line. The line that divides a given line into two equal parts is termed a line bisector.

The appropriate proof is as stated below:

              STATEMENT                                                 REASON

1. AB ≅ FG                                                              Given

2. BF bisects AC and DG                                      Given

3. AB ≅ BC                                                             Definition of a bisector

4. DF ≅ FG                                                             Definition of a bisector

5. AC ≅ DG                                                         Property of two congruent lines

6. BC ≅ DF                                                         Property of two congruent lines

Therefore it can be deduced that BC ≅ DF  (property of two congruent lines)

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A. The mean of the data set is 71.09.

B. The median of the data set is the 12th term which is 74.

C. There is no mode value in the given data set.

D. Median value best represents the data.

A stem leaf method is used to represent discrete data.

The first column are the stem values and the second are the leaf values.

To find the mean of the data set we'll use the weightage average method

which is sum of all the values divided by the number of values.

∴ Mean = 71.09.

Total no. of data is 23 so the median is the 12th term which is 74.

There is no particular frequently occurring data hence no mode.

Median value best represents this data set as in the stem and leaf the most no. of data are in this row.

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Answer: 211200

Step-by-step explanation: Ok so assuming you are asking how many feet you travel per hour going 40 mph. You would travel 211200. There are 5280 ft in a mile. If you multiply that by 40 you get 211200. So you travel 211200ft per hour

Answer:

No

Step-by-step explanation:

Consider the graphs of [tex]y=2^x[/tex] and [tex]y=3^x[/tex].

Answer:

The graph of g(x) = (3x+5) + 8 will be shifted 8 units up from the graph of f(x). Specifically, the graph of g(x) will be 8 units higher than the graph of f(x) for all x.

Step-by-step explanation:

The number of days the lizard that will be taken to reach the top will be 8 days .

Since we're given the information that the spider each day climbs five meters and slides back one meter, that means that it climbs for 4 meters daily .

Number of meters for each day = 5 - 1

= 4 meters

Therefore, since the spider is climbing up a 30 meter building, the number of days required will be :

= 30 / 4

= 15 / 2

= 7.5

= 8 days approximately.

Hence The number of days will be needed to reach the top is 8 days.

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Answer:

x+40= x+x-44

40+44=2x-x

x=84

The four steps to copy and paste involve selecting the text, copying the text, selecting the cursor at destination and pasting the text.

When you copy or cut data from its original place, the computer temporarily stores it in a memory region called the clipboard. Data that has been copied or cut is retrieved from the clipboard and pasted to the new spot when you paste it.

When you copy text, a copy of the original text is made and stored in the clipboard; the original text stays in its original place. The original text is taken out of its original spot and placed in the clipboard when you cut text. The original text is retained when you copy it, but it is removed from its original spot when you cut it. The copied or cut text can be pasted into another area in both situations.

You can follow the following 4 steps to copy and paste text or other items in a computer application:

1. By highlighting it with your pointer, you may choose the text or object that you wish to copy.

2. Right-click the text or object you want to copy, then choose "Copy" from the context menu. As an alternative, you may copy by pressing "Ctrl+C" on your keyboard.

3. Place the cursor where you wish to paste the text or object that was copied.

4. "Paste" may be found in the context menu when you right-click the destination. As an alternative, you may paste by using "Ctrl+V" on your keyboard.

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Answer:

  (i) x = 17 cm

  (ii) one end: 360 cm²; both ends: 720 cm²

  (iii) 14,400 cm³

  (iv) 4000 cm²

Step-by-step explanation:

You want the slant height, base area, volume, and total surface area of a trapezoidal prism with the base isosceles trapezoid having parallel base lengths of 32 and 16, and a height of 15. The distance between bases is 40. All units are cm.

If a center rectangle 16 cm wide and 15 cm high is cut from the base trapezoid, the remaining two triangles have a base of 8 cm and a height of 15 cm. The Pythagorean theorem can be used to find the slant height:

  x² = a² +b²

  x² = 8² +15² = 64 +225 = 289

  x = √289 = 17

The slant height, x, is 17 cm.

The area of the base trapezoid is given by the formula ...

  A = 1/2(b1 +b2)h

  A = 1/2(32 +16)(15) = 360 . . . . square cm

The area of one end surface is 360 cm²; the total area of both end surfaces is 720 cm².

The volume of the prism is the product of the base area and the length of the prism.

  V = Bh

  V = (360 cm²)(40 cm) = 14,400 cm³

The volume of the trapezoidal prism is 14,400 cm³.

The lateral surface area of the prism is the product of the perimeter of the base and the distance between bases.

  LA = Ph

  LA = (32 +16 +2·17 cm)(40 cm) = 3280 cm²

The total surface area is the sum of the lateral area and the area of the two bases:

  SA = LA +2B = (3280 cm²) + 2(360 cm²) = 4000 cm²

The total surface area of the prism is 4000 square centimeters.

__

Additional comment

It appears that the top dashed line in the figure is drawn that way in error. It appears to identify a visible edge, so we expect it to be a solid line.

The median of 2/3 and 4 is 2.33.

The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.

A data set's median value is the point where 50% of the data points have values that are lower or equal to it, and 50% of the data points have values that are higher or equal to it.

To arrange the data points in ascending order for a small data set, count the number of data points (n).

To get the rank of the data point whose value is the median when the number of data points is unequal, multiply the total by 1 and divide the results by 2.

Median = (2/3+4)/2 = 4.66/2 = 2.33

Thus, the median of 2/3 and 4 is 2.33.

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To get the rank of the data point whose value is the median when the number of data points is unequal, multiply the total by 1 and divide the results by 2.

Median = (2/3+4)/2 = 4.66/2 = 2.33

Thus, the median of 2/3 and 4 is 2.33.

In a case whereby least common multiple of two whole numbers is 40. the ratio of the greater number to the lesser number is 5:4.  the two numbers are 10 and 8.

Given as :

The least common multiple of two numbers = LCM = 40

The ratio of the greater number to lesser number = 5 : 4

let the greater number = 5 x

And The smaller number = 4x

∵ The LCM of numbers = 40

So, 5 × 4 × x = 40

Or,  20 × x = 60

x= 3

Then greater number = 5 x = 5 × 2 = 10

And  The smaller number = 4 x = 4 × 2 = 8

Hence The two numbers are  10 and 8

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The y-intercept is (0, 5). Hence, the intercepts of the equation 5x + 3y = 15 are (3,0) and (0,5).

To find the x-intercept we have to substitute y = 0 into the equation, as follows:

9x - 7*0 = -63

9x = -63

Dividing by 9 at both sides of the equation:

9x/9 = -63/9

x = -7

To find the y-intercept we have to substitute x = 0 into the equation, as follows:

9*0 - 7y = -63

-7y = -63

Dividing by -7 at both sides of the equation:

-7y/(-7) = -63/(-7)

y = 9

x-intercept: −7; y-intercept: 9

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The total number of combinations is given by the product between the numbers of options, so there are 8 combinations.

Here we have 3 variables and each can take two values:

x = {x₁, x₂}

y = {y₁, y₂}

z = {z₁, z₂}

The total number of combinations is equal to the product between the number of options that we have for each variable, then the number of combinations is:

C = 2*2*2 = 4*2 = 8

C = 8

There are 8 combinations.

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Answer: $7.65 for a dozen/12

Step-by-step explanation: 2.55 x 3

The solution of the differential equation would be [tex]r = \dfrac{(\theta - cos\theta) + C}{(sec\theta + tan\theta)}[/tex].

Using the concept of integration that states,

Integration in mathematics is a technique for integrating or adding up the parts to get the total. It involves a differentiation process in reverse.

Given that,

The differential equation is,

[tex]\dfrac{dr}{d\theta } + r sec(\theta ) = cos (\theta )[/tex]

Hence the auxiliary solution,

[tex]\mu (\theta ) = e^{\int\limits {sec\theta} \, d\theta}[/tex]

[tex]\mu (\theta ) = e^{ln |\limits {sec\theta + tan\theta|[/tex]

[tex]\mu (\theta ) = |sec\theta + tan\theta|[/tex]

Hence multiply both sides by [tex]\mu(\theta)[/tex],

[tex]({sec\theta + tan\theta)\dfrac{dr}{d\theta } + ({sec\theta + tan\theta) r sec(\theta ) = cos (\theta ) ({sec\theta + tan\theta)[/tex]

[tex]\dfrac{d}{d\theta} (r \times (sec\theta + tan\theta) = (1 + sin\theta)[/tex]

Integration on both sides,

[tex]r \times (sec\theta + tan\theta) = (\theta - cos\theta) + C[/tex]

Therefore, the solution is,

[tex]r = \dfrac{(\theta - cos\theta) + C}{(sec\theta + tan\theta)}[/tex]

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