what is factoring: problems involving factors of polynomials

The limit of the given function as x approaches infinity is infinity.

To find the limit of the given function as x approaches infinity, we need to analyze the behavior of the function for very large values of x.

First, we can observe that the term 49x^2 dominates the other terms in the function as x gets very large. This is because the square of any number grows much faster than a linear function (x) or a constant function (-7x).

Therefore, we can simplify the given function as follows:

lim x→[infinity] (49x^2 + x − 7x) = lim x→[infinity] 49x^2

Now we can directly evaluate the limit by applying the rule that states if a function is of the form ax^n where a and n are constants and x approachesinfinity, then the limit is infinity if n > 0 and negative infinity if n < 0.

In this case, a = 49 and n = 2, which is greater than 0. Therefore, the limit as x approaches infinity is infinity.

To know more about function visit:-

https://brainly.com/question/12431044

#SPJ11

Answer:

Step-by-step explanation:

1).

(a). Yes, the graph is linear.

(b). Rate of change is a slope m, m = (60 - 80) / (10 - 0) = - 2

2). Graph is not proportional, because y-intercept is (0, 80) and not (0, 0)

Answer:

7 / 69

Step-by-step explanation:

Rabbit = 8

Mice = 12

Snake = 4

Total = (8 + 12 + 4) = 24

If each animal has equal probability of being caught ;

Probability of catching a rabbit ;

Number of rabbits / total number of animals

P(catching 1 rabbit) = 8 / 24 = 1/3

P(catching 2nd rabbit) :

Number of rabbits left = 8 - 1 = 7

Total number of rabbits = 24 - 1 = 23

Hence, P(catching 2nd rabbit) = 7 /23

Probability of catching two rabbits :

1 /3 * 7/ 23 = 7 / 69

The pair of 1st line is vertical opposite angle and 2nd one is adjacent angles.

What is vertical opposite angle ?

When 2 lines meet one another, then the other angles, shaped thanks to intersection ar known as vertical angles or vertically opposite angles. A try of vertically opposite angles ar continuously adequate to one another. Also, a angle and its adjacent angle are supplementary angles, i.e., they add up to a hundred and eighty degrees.

Main body:

1st pair of line intersect each other , so ∠1 = ∠2 by using vertical opposite angle.

2nd pair of line also intersect each other , so ∠3 ,∠4 are adjacent angles.

Hence ,pair of 1st line is vertical opposite angle and 2nd one is adjacent angles.

To know more about vertical opposite angle , click on the link below

https://brainly.com/question/26768295

#SPJ1

Answer:

Step-by-step explanation:

The value of cos(x/2) is required.

The required value is \(\cos(x/2)=-\sqrt{\dfrac{5-2\sqrt{5}}{10}}\)

The given value is

\(\tan x=\dfrac{1}{2}\) where x is in quadrant III.

From trigonometric identities we have

\(\sec^2x=1+\tan^2x\\\Rightarrow \sec^2x=1+(\dfrac{1}{2})^2\\\Rightarrow \sec^2x=\dfrac{5}{4}\\\Rightarrow \cos^{2}x=\dfrac{4}{5}\\\Rightarrow \cos x=\dfrac{2}{\sqrt{5}}\)

If x lies in quadrant III then x/2 lies in quadrant II.

Cos is negative in quadrant III so,

\(\cos x=-\dfrac{2}{\sqrt{5}}=-\dfrac{2\sqrt{5}}{5}\)

\(\cos \dfrac{x}{2}=\pm \sqrt{\dfrac{1+\cos x}{2}}\\ =\pm\sqrt{\dfrac{1-\dfrac{2\sqrt{5}}{5}}{2}}\\ =\pm\sqrt{\dfrac{5-2\sqrt{5}}{10}}\)

Since, x/2 lies in the second quadrant and cos (x/2) is negative.

The required value is \(-\sqrt{\dfrac{5-2\sqrt{5}}{10}}\)

Learn more about trigonometry:

https://brainly.com/question/24349828

The correct option is that in science, a theory encompasses many hypotheses.

In science, a theory is a well-established explanation that is supported by scientific evidence and is extensively tested through experimentation and observation.

It's a big idea that includes several connected hypothese that have been thoroughly investigated and found to be true. A theory that hasn't been put to the test is called a hypothesis.

A hypothesis is a proposed explanation that is subject to further testing. It's an untested hypothesis that hasn't yet undergone experimental validation. Theories are the result of a drawn-out and labor-intensive process of experimentation and observation.

They can never be established as an absolute answer to a scientific problem. Instead, they lay the groundwork for additional investigation and experimentation in that field of study.

Know more about hypothesis here:

https://brainly.com/question/29576929

#SPJ11

Because there is no pattern in the graph, we conclude that there is no correlation between the two variables, thus, the correct option is C.

In the horizontal axis, we can see the pounds of fruit bought, while on the vertical axis we can see the pounds of vegetables bought.

If you see the graph, you can see that there is no pattern. Meaning that there is no relation between these two quantities.

So there is no correlation between the variables "pounds of fruit bought" and "pounds of vegetables bought.".

Then the correct option is C.

"There is no association between the pounds of fruit purchased and the pounds of vegetables purchased"

If you want to learn more about correlation, you can read:

https://brainly.com/question/14289293

Answer:

The observation I can make for the values of pi for circles A and B is that the value of pi remains the same whether we find pi using the circumference of the circle or the Area of the circle, the value of pi remains the same for both circles.

Pi = π = 3.14

Step-by-step explanation:

Part A: Using the formula for circumference, solve for the value of pi for each circle. (4 points)

The formula for the circumference of circle when Diameter is given = πD

π = Circumference / Diameter

For Circle A :

Circle A has a diameter of 7 inches, a circumference of 21.98 inches.

π = 21.98 inches/7 inches

π = 3.14

For Circle B

The diameter of circle B is 6 inches, the circumference is 18.84 inches

π = 18.84 inches/6 inches

π = 3.14

Part B: Use the formula for area and solve for the value of pi for each circle. (4 points)

The formula for the area of the circle = πr²

Circle A has a diameter of 7 inches, an area of 38.465 square inches.

r = Radius = 7 inches ÷ 2

= 3.5 inches

π = Area / Radius²

π = 38.465 in²/(3.5 inches)²

π = 3.14

For Circle B

The diameter of circle B is 6 inches, and the area is 28.26 square inches.

r = Radius = 6 inches ÷ 2

= 3 inches

π = Area / Radius²

π = 28.26 in²/(3 inches)²

π = 3.14

Part C: What observation can you make about the value of pi for circles A and B? (2 points)

The observation I can make for the values of pi for circles A and B is that the value of pi remains the same whether we find pi using the circumference of the circle or the Area of the circle, the value of pi remains the same for both circles.

The observation is that the values of \(\pi\) remain 3.14 when \(\pi\) is calculated from the areas and circumferences of both circles

The given parameters are:

Circle A

Circle B

The circumference of a circle is:

\(C= \pi d\)

Make \(\pi\) the subject

\(\pi = \frac{C}{d}\)

For circle A, we have:

\(\pi_A = \frac{21.98}{7}\)

\(\pi_A = 3.14\)

For circle B, we have:

\(\pi_B = \frac{18.84}{6}\)

\(\pi_B = 3.14\)

Hence, the value of \(\pi\) in both circles is 3.14

The given parameters are:

Circle A

Circle B

The area of a circle is:

\(A= \pi \frac{d^2}{4}\)

Make \(\pi\) the subject

\(\pi = \frac{4A}{d^2}\)

For circle A, we have:

\(\pi_A = \frac{4 * 38.465 }{7^2}\)

\(\pi_A = 3.14\)

For circle B, we have:

\(\pi_B = \frac{4 * 28.26 }{6^2}\)

\(\pi_B = 3.14\)

Hence, the value of \(\pi\) in both circles is 3.14

The observation is that the values of \(\pi\) in both circles when area and circumference are used is 3.14

Read more about area and circumference at:

https://brainly.com/question/15673093

Answer:

Slope is 3/5

Step-by-step explanation:

Answer:

Its AA similaroty theorem

The margin of error (E) is  0.399 minutes.

The correct option is (B).

The margin of error can be defined as the difference between the value of the true population parameter and the observed value. If the sample size is large, then the margin of error will be small and if the sample size is small, then the margin of error will be large.

The given information from the question is:

The sample size is n = 600

The population standard deviation is  σ = 4.5 minutes,

The confidence level is CL = 0.97

The formula to calculate the margin of error (E) is

\(E=\frac{Z^*(\sigma)}{\sqrt{n} }\)

The level of significance \((\alpha )\) is:

\(\alpha =1-CL\)

   = 1 - 0.97

   = 0.03

At the significance level 0.03, the two tailed critical value \((Z^*)\) obtained from the standard normal table is : ±2.17

Substituting values, the margin of error (E) can be calculated as:

\(E=\frac{Z^*(\sigma)}{\sqrt{n} }\)

\(E=\frac{2.17(4.5)}{\sqrt{600} }\)

E = 0.399 minutes

Hence, The margin of error (E) is  0.399 minutes.

Learn more about Margin of error at:

https://brainly.com/question/31764430

#SPJ4

The effect of educational attainment on the time spent sleeping per week is examined using the simple regression model sleep = β0 + β1educ + u. The sample mean and standard deviation of sleep and educ can be calculated using the data in PS2 Q4.dta.

To find the sample mean of sleep and educ, sum up all the values of sleep and educ respectively and divide by the total number of observations.

The standard deviation can be calculated by taking the square root of the variance, where the variance is the average of the squared deviations from the mean.

These calculations provide measures of central tendency and dispersion for the variables sleep and educ.

Learn more about educational attainment

brainly.com/question/28111346

#SPJ11

the owner used 10 pounds of chocolates and 10 pounds of lollipops to make 20 pounds of candy.

Let's assume the owner used x pounds of chocolates and y pounds of lollipops to make a total of 20 pounds of candy.

The cost of the chocolates is $5.25 per pound, so the cost of x pounds of chocolates is 5.25x dollars.

The cost of the lollipops is $1.25 per pound, so the cost of y pounds of lollipops is 1.25y dollars.

The total cost of the candy is $3.25 per pound, so the cost of 20 pounds of candy is 3.25 * 20 = 65 dollars.

Since the total cost of the candy is the sum of the costs of the chocolates and lollipops, we can write the equation:

5.25x + 1.25y = 65

We also know that the total weight of the candy is 20 pounds, so we have another equation:

x + y = 20

Now we have a system of two equations:

5.25x + 1.25y = 65

x + y = 20

We can solve this system of equations to find the values of x and y.

Multiplying the second equation by 1.25, we get:

1.25x + 1.25y = 25

Now, we can subtract this equation from the first equation to eliminate the y variable:

5.25x + 1.25y - (1.25x + 1.25y) = 65 - 25

4x = 40

Dividing both sides by 4, we find:

x = 10

Substituting the value of x into the second equation:

10 + y = 20

y = 20 - 10

y = 10

Therefore, the owner used 10 pounds of chocolates and 10 pounds of lollipops to make 20 pounds of candy.

Learn more about Linear equation here

https://brainly.com/question/32634451

#SPJ4

When the distance between the objects is 5.5 meters, the force (F) is approximately 0.0827 Newtons (rounded to 2 decimal places).

According to the given information, the force (F) between two objects is inversely proportional to the square of the distance (d) between them. Mathematically, this can be represented as \(F = k/d^2,\)

where k is the constant of proportionality.

To find the value of k, we can use the given data point where the force is 0.07 Newtons when the distance is 6 meters. Plugging these values into the equation, we get:

\(0.07 = k/6^2\)

Simplifying, we have:

0.07 = k/36

To solve for k, we can multiply both sides of the equation by 36:

\(k = 0.07 \times 36\)

k = 2.52

Now that we have the value of k, we can calculate the force (F) when the distance (d) is 5.5 meters:

\(F = 2.52 / 5.5^2\)

F ≈ 0.0827

For similar question on distance.

https://brainly.com/question/30339968

#SPJ11

Answer:

Andre     Noah     Weeks

 20           10             1

 25         17.50         2

 30           25            3

Step-by-step explanation:

The set of parametric equations for the rectangular equation y = 3x - 9, with t = 0 at the point (4, 3), is y = 3t + 3.

To find a set of parametric equations that satisfy the given condition y = 3x - 9, t = 0 at the point (4, 3), we can express the rectangular equation in parametric form using a parameter, typically denoted by t.

Let's begin by introducing the parameter t and assigning initial values for x and y at t = 0. From the given condition, we have x = 4 and y = 3 when t = 0.

Now, we can express x and y in terms of t and write the parametric equations:

x = f(t)

y = g(t)

To find the expressions for f(t) and g(t), let's analyze the relationship between x and y in the rectangular equation y = 3x - 9.

From the equation, we can rearrange it to solve for x:

x = (y + 9) / 3

Now, we have an expression for x in terms of y. However, we want to express x and y in terms of the parameter t. To do this, we substitute y in terms of t into the expression for x:

x = ((g(t) + 9) / 3)

Therefore, we have the parametric equation:

x = ((g(t) + 9) / 3)

Next, we need to determine the expression for g(t). To find g(t), we observe that when t = 0, y = 3. This means that g(0) = 3. Since the slope of the equation y = 3x - 9 is 3, we can express g(t) as:

g(t) = 3t + 3

Substituting this expression for g(t) into the equation for x, we get:

x = ((3t + 3 + 9) / 3)

x = (3t + 12) / 3

x = t + 4

Therefore, the set of parametric equations for the rectangular equation y = 3x - 9, with t = 0 at the point (4, 3), is:

x = t + 4

y = 3t + 3

These parametric equations represent the relationship between x and y in terms of the parameter t. As t varies, the point (x, y) traces out a curve on the Cartesian plane. In this case, the curve is a straight line with a slope of 3 and passing through the point (4, 3). As t increases or decreases, the point moves along this line, resulting in a linear relationship between x and y.

Learn more about parametric equations here

https://brainly.com/question/30451972

#SPJ11

Answer:

<12

Step-by-step explanation:

If you look at the shape each angle makes, 12 is different than 1, the rest are the same.

Congruent means to be the same.

12 is the Only one that is NOT the same.

Hope this helps!

The solution to the system of equations graphed below is,

⇒ (0, 1)

Since, We have to given that;

Two system of equations are,

⇒ g (x) = 3x + 2

⇒ f (x) = |x - 1| + 1

Here, The graph of both system of equation are shown in graph.

We know that;

In a graph, the solution of system of equation are represented by a intersection point of both graph.

Here, In the graph of system of equation,

Intersection point is,

⇒ (0, 1)

Hence, The solution to the system of equations graphed below is,

⇒ (0, 1)

Learn more about the mathematical expression visit:

brainly.com/question/1859113

#SPJ1

Answer:

4 units to the right

Step-by-step explanation:

(x,y)

so to go from -2 to 2 you must move 4 spaces.

On the x axis that is to the right.

Answer:

Step-by-step explanation:

Total man hours:

Required days:

Answer:

You are gonna need 4 cups of milk.

Step-by-step explanation:

1/3 for 10 cookies so x 3= 1 cup for 30 cookies. 1 cup and 30 cookies x 4 = 4 cups and 120 cookies

The function f(x₁, x₂) = x₁x₂ on R²+ is convex, while the function f(x₁, x₂) = x₁/x₂ on R² is neither convex nor concave.

To determine the convexity of a function, we need to examine the Hessian matrix.

The Hessian matrix of a function consists of its second-order partial derivatives. For the function f(x₁, x₂) = x₁x₂ on R²+, the Hessian matrix is:

H = [0 1]

[1 0]

To determine if the function is convex, we need to check if the Hessian matrix is positive semidefinite (all eigenvalues are nonnegative). In this case, the eigenvalues of the Hessian matrix are both nonnegative, indicating that the function is convex.

On the other hand, for the function f(x₁, x₂) = x₁/x₂ on R², the Hessian matrix is:

H = [0 -1/x₂²]

[-1/x₂² 2x₁/x₂³]

To determine convexity, we need to check the eigenvalues of the Hessian matrix. However, the eigenvalues of the Hessian matrix are dependent on the values of x₁ and x₂. For instance, if x₂ = 0, the Hessian matrix becomes undefined.

Since the function f(x₁, x₂) = x₁/x₂ does not have a constant Hessian matrix, we cannot conclude its convexity. Therefore, the function is neither convex nor concave.

In conclusion, the function f(x₁, x₂) = x₁x₂ on R²+ is convex, while the function f(x₁, x₂) = x₁/x₂ on R² is neither convex nor concave due to its variable Hessian matrix.

Learn more about  Hessian matrix :

https://brainly.com/question/32547012

#SPJ11

The shape of the cross section of a triangular pyramid sliced by a plane that is perpendicular to its base is triangle.

The non-empty intersection of a solid body in three dimensions with a plane, or its equivalent in higher dimensions, is referred to as a cross section in geometry and science. Many parallel cross-sections are produced when an object is sliced. The boundary of a cross-section in three dimensions that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line.

A triangle-shaped pyramid essentially has a triangle-shaped base. The cross section formed when an imaginary line is produced to cut through perpendicular to the base will still have a triangular orientation, but it will be smaller than it was when the pyramid's original triangular orientation or sides were there.

Therefore, the shape of the cross section of a triangular pyramid sliced by a plane that is perpendicular to its base is triangle.

To know more about triangular pyramid, visit:

https://brainly.com/question/24046619

#SPJ4

The probability of rolling a score of 2 is 1/16, the probability of rolling a score of 3 or 7 is 1/8, the probability of rolling a score of 4 or 6 is 3/16, and the probability of rolling a score of 5 is 1/4. This is the probability distribution for the score obtained when rolling two tetrahedral dice.

To create a probability distribution for the score obtained by rolling two tetrahedral dice, we need to calculate the probability of each possible score that can be obtained by adding the numbers on the bottom faces of the two dice.

There are 16 possible outcomes when rolling two tetrahedral dice, since each die has 4 faces and there are 4 * 4 = 16 possible combinations of faces that can be rolled. To calculate the probability of each possible outcome, we can use the following steps:

List all the possible outcomes of rolling two tetrahedral dice and add up the numbers on the bottom faces to determine the score obtained.

Here are all 16 possible outcomes, along with the sum of the numbers on the bottom faces (which is the score obtained):

(1,1) = 2

(1,2) = 3

(1,3) = 4

(1,4) = 5

(2,1) = 3

(2,2) = 4

(2,3) = 5

(2,4) = 6

(3,1) = 4

(3,2) = 5

(3,3) = 6

(3,4) = 7

(4,1) = 5

(4,2) = 6

(4,3) = 7

(4,4) = 8

Calculate the probability of each possible score by counting the number of outcomes that result in that score, and dividing by the total number of possible outcomes.

For example, to calculate the probability of a score of 2, we count the number of outcomes that result in a sum of 2, which is only one: (1,1). Since there are 16 possible outcomes in total, the probability of rolling a score of 2 is 1/16.

We can repeat this process for each possible score to create the following probability distribution:

Score Probability

2 1/16

3 2/16 = 1/8

4 3/16

5 4/16 = 1/4

6 3/16

7 2/16 = 1/8

8 1/16

So the probability of rolling a score of 2 is 1/16, the probability of rolling a score of 3 or 7 is 1/8, the probability of rolling a score of 4 or 6 is 3/16, and the probability of rolling a score of 5 is 1/4. This is the probability distribution for the score obtained when rolling two tetrahedral dice.

Learn more about tetrahedral dice.

brainly.com/question/2563730

#SPJ11

The area of the triangle in simplest radical form is 20√3 square units.

In Mathematics and Geometry, the area of a triangle can be calculated by using the following mathematical equation (formula):

Area of triangle = 1/2 × b × h

Where:

By substituting the given side lengths into the formula for the area of a triangle, we have the following;

Area of triangle ABC = 1/2 × b × h

Area of triangle ABC = 1/2 × 5√2 × 4√6

Area of triangle ABC = 5√2 × 2√6

Area of triangle ABC = 20√3 square units.

Read more on area of triangle here: brainly.com/question/12548135

#SPJ1

Complete Question:

In the accompanying diagram of ABC, altitude BD = 4√6 and AC = 5√2. Find the area of the triangle in simplest radical form.

Answer:  

m<1 = 51

m<2 = 34

m<3 = 95

m<4 = 38

m<5 = 47

m<6 = 74

m<7 = 59

Step-by-step explanation: because to get angle 1 you take 180 (because its is a straight line) and subtract the angle 129 giving you m<1 witch is 51. the second one is 180 - (m<1 + 95) =m<2 witch is 34 so m<2 = 34. third is the angle 95 because the lines that intersect give equal opposite angles so m<3 = 95. Four is 180 - (m<3 + 47) =m<4 witch is 38. fifth is opposite angle so it is 47. sixth is m<5 +m<6 + m<7 = 180 so m<5 is 47 and we can find m>7 by subtracting from 180 the angle 121 giving us m<7 = 59 so we will add m<5 and m<7 and subtract from 180 giving us m<6 witch is 74.    

Answer: 6 hours washing cars and 4 hours tutoring

Step-by-step explanation:

# of hours washing cars can be represented as x

# of hours tutoring can be represented by y

10x + 12y = $108

y + 2 = x

10 (y +2) + 12y = 108

10y + 20 + 12y = 108

22y + 20 = 108

22y = 88

y = 4 hrs tutoring

4 + 2 = 6 hrs washing cars

Check:

10(6) + 4(12) = 108

108 = 108

The number of hours Naomi worked washing cars last week and the number of hours she worked tutoring last week will be 4 and 6 hours respectively.

An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.

It is given that, Naomi is earning $10 per hour washing cars and $12 per hour tutoring during the summer. Naomi worked 2 more hours washing cars last week than tutoring, earning a total of $108.

Suppose the number of hours invested in washing cars is x and tutoring be y.

It is given that, Naomi is earning $10 per hour washing cars and $12 per hour tutoring during the summer and earning a total of $108.The obtained equation is

10x + 12y = $108 -----(1)

If Naomi worked 2 more hours washing cars last week than tutoring. The obtained equation is

y + 2 = x -----(2)

Substitute the value of y in equation 1

10 (y +2) + 12y = 108

10y + 20 + 12y = 108

22y + 20 = 108

22y = 88

y = 4 hrs

The value of x is,

x=y+2

x=4+2

x=6

Thus, the number of hours Naomi worked washing cars last week and the number of hours she worked tutoring last week will be 4 and 6 hours respectively.

Learn more about the equation here,

https://brainly.com/question/10413253

#SPJ2

To set up the equation system in matrix format, we can rewrite the given system of equations using matrix notation.

Let's define the following matrices and vectors:

- Y: Output vector (Y = [Y1, Y2, Y3]ᵀ)

- Q: Endowment vector (Q = [Q1, Q2, Q3]ᵀ)

- S: Input-output matrix (S = [sij])

Now, let's expand and rewrite the equation system using matrix notation:

∑(j) sij Yj = Qi

Expanding this equation for each i, we get:

s11Y1 + s12Y2 + s13Y3 = Q1

s21Y1 + s22Y2 + s23Y3 = Q2

s31Y1 + s32Y2 + s33Y3 = Q3

We can rewrite the above system of equations in matrix format as:

S * Y = Q

where S is the input-output matrix, Y is the output vector, and Q is the endowment vector.

In matrix format, the equation system can be written as:

⎡ s11  s12  s13 ⎤  ⎡ Y1 ⎤   ⎡ Q1 ⎤

⎢ s21  s22  s23 ⎥  ⎢ Y2 ⎥ = ⎢ Q2 ⎥

⎣ s31  s32  s33 ⎦  ⎣ Y3 ⎦   ⎣ Q3 ⎦

This is the equation system represented in matrix format.

learn more about matrix here:

https://brainly.com/question/29132693

#SPJ11

There could be many other combinations of scores that would result in the same mean, median, and mode.

What is mean ?

Mean can be defined as ratio sum of given observations, total number of observations.

Since the median is 7 and the mode is 8, we can assume that the majority of her scores were in the range of 7-8 points. We can also assume that there were a few games in which she scored lower than 7 points and a few games in which she scored higher than 8 points.

Here's one possible set of scores that would result in a mean of 6, a median of 7, and a mode of 8:

Game 1: 3 points

Game 2: 5 points

Game 3: 6 points

Game 4: 7 points

Game 5: 7 points

Game 6: 8 points

Game 7: 8 points

Game 8: 8 points

Game 9: 9 points

Game 10: 10 points

Game 11: 11 points

Note that this is just one possible set of scores that would fit the given statistics.

Therefore, There could be many other combinations of scores that would result in the same mean, median, and mode.

To learn more about Mean from given link.

https://brainly.com/question/21533310

#SPJ1