Answer:
7
Step-by-step explanation:
To find the mean of a set of numbers, we add up all the numbers and divide the sum by the total number of numbers.
For the given sample, we have:
10 + 6 + 8 + 6 + 5 = 35
There are 5 numbers in the sample, so we divide the sum by 5:
35 / 5 = 7
Therefore, the mean of the sample is 7.
find invertible matrices such that is non-invertible. choose so that (1) neither is a diagonal matrix and (2) are not scalar multiples of each other.
Invertible matrices P = [1 -2; 0 1] and Q = [1 0; 2 1] such that
A = PQ = [1 2; 2 -3] is non-invertible.
To find invertible matrices such that a given matrix is non-invertible, we can use the fact that if A is non-invertible, then the system of linear equations Ax = 0 has a non-trivial solution. This means that there exists a non-zero vector x such that Ax = 0.
Let's start with a non-invertible matrix A, for example:
A = [1 2; 2 4]
The determinant of A is 0, which means that A is non-invertible.
To find a non-zero vector x such that Ax = 0,
We can solve the system of linear equations:
x + 2y = 0
2x + 4y = 0
This system is equivalent to the single equation:
x + 2y = 0
If we choose y = 1, then x = -2, and we get the non-zero vector:
x = [-2; 1]
Now we can use x to construct invertible matrices P and Q such that
PQ = A, as follows:
P = [1 -2; 0 1]
Q = [1 0; 2 1]
The inverse of P is:
P^-1 = [1 2; 0 1]
And the inverse of Q is:
Q^-1 = [1 0; -2 1]
We can verify that P and Q are invertible and that PQ = A:
PQ = [1 -2; 0 1][1 0; 2 1]
PQ = [1 -2; 2 -4 + 1]
PQ = [1 2; 2 -3] = A
Therefore, we have found invertible matrices P and Q such that A = PQ is non-invertible.
Note:- that neither P nor Q is a diagonal matrix, and they are not scalar multiples of each other.
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The complete question may be:
Find non-invertible matrices A, B such that A+B is invertible. Choose
A, B, so that (1) neither is a diagonal matrix and (2) A, B are not scalar multiples of each other.
What are the features of the quadratic function graphed in the figure?
A) Vertex = (–3,4), y-intercept = (0,–5), x-intercepts = (–5,0) and (–1,0), axis of symmetry is x = –3
B) Vertex = (3,–4), y-intercepts = (–1,0) and (–5,0), x-intercept = (0,5), axis of symmetry is x = 3
C) Vertex = (–3,4), y-intercept = (0,–5), x-intercepts = (1,0) and (5,0), axis of symmetry is x = –3
D)Vertex = (–4,3), y-intercept = (5,0), x-intercepts = (0,1) and (0,5), axis of symmetry is x = –4
Answer:
A) Vertex = (-3, 4), y-intercept = (0, -5), x-intercepts = (-5, 0) and (-1, 0), axis of symmetry is x = -3
Step-by-step explanation:
The vertex of a quadratic function is the turning point. As this parabola opens downwards, the vertex is the maximum point of the graph. From inspection of the graph, the maximum point is at (-3, 4). Therefore:
The vertex of the quadratic function is (-3, 4).The y-intercept is the point at which the curve crosses the y-axis. From inspection of the graph, the curve crosses the y-axis at y = -5. Therefore:
The y-intercept of the quadratic function is (0, -5).The x-intercepts are the points at which the curve crosses the x-axis. From inspection of the graph, the curve crosses the x-axis at x = -5 and x = -1. Therefore:
The x-intercepts of the quadratic function are (-5, 0) and (-1, 0).The axis of symmetry is the vertical line that passes through the vertex of the parabola so that the left and right sides of the parabola are symmetrical. So the axis of symmetry is the x-value of the vertex. Therefore:
The axis of symmetry of the quadratic function is x = -3.Estimate the sum of 202 and 57.
Answer:
259
Step-by-step explanation:
Answer:259
Step-by-step explanation:
202+57=259
Find the inequality represented by the graph.
Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
y=ln(x), y=0,x=5; about the y -axis___ dx
The integral for the volume of the solid is ∫2π(eᵇ)(dy)
In calculus, finding the volume of a solid is an important concept. One method to do this is by using integration.
To find the volume of the solid obtained by rotating the region bounded by y=ln(x), y=0, and x=5 about the y-axis, we can use the method of cylindrical shells. This involves dividing the region into infinitely thin vertical strips and rotating each strip around the y-axis to create cylindrical shells.
To set up the integral, we can use the formula for the volume of a cylindrical shell, which is 2πrhΔx, where r is the distance from the y-axis to the edge of the strip, h is the height of the strip, and Δx is the thickness of the strip.
Since we are rotating around the y-axis, we need to express the curves in terms of y instead of x. Using the inverse of the natural logarithm function, we can rewrite
=> y=ln(x) as x=eᵇ.
Thus, the region is bounded by x=5, y=0, and
=> x=eᵇ.
Next, we need to find the distance from the y-axis to the edge of the strip, which is simply the x-value at a given y. This is given by r=e^y.
The height of the strip is simply Δy, the thickness of the strip. To find this, we can take the difference between two consecutive y-values. Thus, Δy=dy.
Putting it all together, the integral for the volume of the solid is:
=> ∫2π(eᵇ)(dy)
This integral represents the sum of the volumes of all the cylindrical shells that make up the solid. However, we still need to evaluate the integral to get the actual volume of the solid.
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A local fish market is selling fish and lobsters by the pound. The fish, f, costs $6.25 a pound, while the lobster, l, costs $9.50 a pound. The fish market sells 20.5 pounds and makes $149.25. How many pounds of fish and pounds of lobster did the market sell?
Enter a system of equations to represent the situation, then solve the system.
The system of equations to represent the situation are f + l = 20.5 and 6.25f + 9.5l = 149.25, solution is 14 pounds and 6.5 pounds
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
We are given that;
Cost of fish=$6.25
Cost of lobster=$9.50
Cost of 20.5 pounds =$149.25
Now, let's use the variables f and l to represent the pounds of fish and lobster sold, respectively. We can set up two equations based on the information given:
f + l = 20.5 (the total weight sold is 20.5 pounds)
6.25f + 9.5l = 149.25 (the total revenue made is $149.25)
Now we can solve the system of equations using any method we prefer. Here's one way to do it using substitution:
f + l = 20.5 (equation 1)
f = 20.5 - l (solve equation 1 for f)
6.25f + 9.5l = 149.25 (equation 2)
6.25(20.5 - l) + 9.5l = 149.25 (substitute f = 20.5 - l into equation 2)
128.13 - 6.25l + 9.5l = 149.25 (distribute the 6.25)
3.25l = 21.12 (combine like terms)
l = 6.5 (divide both sides by 3.25)
Now we can use this value of l to find f:
f + 6.5 = 20.5 (from equation 1)
f = 14 (subtract 6.5 from both sides)
Therefore, by the equations the fish market sold 14 pounds of fish and 6.5 pounds of lobster.
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Two employees of a company have to file expense reports for their travel. The first had taken 7 trips with a mean trip cost of $750. The second took a total of 5 trips with a mean cost of $1200. The accounting department wants to determine the mean cost of all the trips that these two employees took. What is the mean cost of all 12 trips? Proposed Solution: Since we have two costs (one for each employee) we can find the mean cost of both employees by the following computation: (750 + 1200) / 2 = 975 What was done wrong in the proposed solution? A. Since there are 12 total trips, the calculation should be (750 + 1200) / 12 B. Since there are 7 trips with a mean of 750 and 5 with a mean of 1200, the calculation should be (7-750 + 5-1200) / 12 C. Since we do not have the original data from each of the 12 trip costs, a mean cost cannot be calculated. D. Since the two costs are not close to one another, a median should be used. E. The proposed calculation is correct.
The correct answer is A, as the mean cost of all 12 trips can be calculated by adding up the total cost of all the trips and dividing by the total number of trips.
The proposed solution is wrong as the mean cost of all 12 trips cannot be calculated by taking the average of the means of the two employees. Taking the average of the means of the two employees assumes that they took the same number of trips and that their trip costs are equally weighted in the total cost calculation, which is not necessarily true.
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How are the zeros of the polynomial function p(x)=(2x−1)(5x+7)(8x+9)
related to the coefficients of p(x)
written in standard form?
Answer:
Step-by-step explanation:
The zeros of a polynomial function are the values of x for which the function equals zero. For the given polynomial function p(x) = (2x - 1)(5x + 7)(8x + 9), the zeros can be found by setting each factor equal to zero and solving for x:
2x - 1 = 0, x = 1/2
5x + 7 = 0, x = -7/5
8x + 9 = 0, x = -9/8
Therefore, the zeros of the function are x = 1/2, x = -7/5, and x = -9/8.
In standard form, the polynomial function p(x) can be written as:
p(x) = 80x^3 + 122x^2 - 127x - 63
The relationship between the zeros of p(x) and its coefficients can be seen in Vieta's formulas. Vieta's formulas state that for a polynomial function of degree n with roots r1, r2, ..., rn, the coefficients of the polynomial can be expressed as:
a0 = (-1)^n * p0
a1 = (-1)^(n-1) * p1 / p0
a2 = (-1)^(n-2) * p2 / p0
...
an-1 = (-1) * pn-1 / p0
an = pn / p0
where p0 is the coefficient of the highest degree term (the leading coefficient), and the pi are the elementary symmetric polynomials, which are given by:
p1 = r1 + r2 + ... + rn
p2 = r1r2 + r1r3 + ... + rn-1rn
...
pn-1 = r1r2...rn-1 + r1r2...rn-2 + ... + rn-2rn-1
pn = r1r2...rn
Using Vieta's formulas, we can see that for the polynomial function p(x) given above, the coefficients are related to the zeros as follows:
a0 = -63
a1 = -127
a2 = 122
a3 = 80
And we can also see that:
a0 = (-1)^3 * p0 = -63
a1 = (-1)^(3-1) * p1 / p0 = -127
a2 = (-1)^(3-2) * p2 / p0 = 122
a3 = (-1)^(3-3) * p3 / p0 = 80
Therefore, the coefficients of the polynomial are related to the zeros through Vieta's formulas, which express the coefficients as functions of the zeros, and vice versa.
Communicate and Justify
A store made $650 on Monday. It made $233 on Tuesday
morning and $378 on Tuesday afternoon.
Leah says the store made more money on Tuesday.
Her work is shown at the right.
1. What is Leah's argument? How does she support it?
2. Tell how you can analyze Leah's reasoning.
3. Does Leah's reasoning make sense?
Leah's argument is that the rounded up figures for Tuesday sales are greater than the sales for Monday. She supports it by summing up the sales figures.
Leah's reasoning is wrong because she rounded up $ 233 to $ 300 instead of to $ 200.
Leah's reasoning therefore does not make sense.
What should Leah have done ?Leah attempts to round the Tuesday sales figures to the nearest 100. In doing so, she rounded $ 233 to $ 300 instead of $ 200 which was the closest.
If she had done so, the result would be :
= 200 + 400
= $ 600
This would then show that Monday's figures were higher than Tuesday.
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Solve the system of equations using the linear combination method.
-4x - 2y = 26
-
-5x – 2y = 35
-
Enter your answers in the boxes.
X =
y =
The value of x is 9.
The value of y is 5.
What is an equation?An equation contains one or more terms with variables connected by an equal sign.
Example:
2x + 4y = 9 is an equation.
2x = 8 is an equation.
We have,
4x - 2y = 26
This can be written as,
4x - 26 = 2y ______(1)
5x – 2y = 35
This can be written as,
5x - 35 = 2y _______(2)
From (1) and (2),
4x - 26 = 5x - 35
35 - 26 = 5x - 4x
9 = x
x = 9
And,
Substituting x = 9 in (1),
4x - 26 = 2y
4 x 9 - 26 = 2y
36 - 26 = 2y
2y = 10
y = 5
Thus,
The solution is (9, 5).
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The entire high school marching band is going on a field trip. According to school district policy, the number of students on the trip determines how many parent chaperones are required to go along. s = the number of band students on the trip p = the number of parent chaperones required on the trip Which variable is the dependent variable?
The number of parent chaperones required on the trip (p) variable is the dependent variable.
According to the question,
s = the number of band students on the trip
p = the number of parent chaperones required on the trip
As the question already said the number of students on the trip determines how many parent chaperones are required to go along, therefore s , the variable of number of student is a independent variable as it does not depends on another variable.
And 'p' , the number of parent chaperones required on the trip depend on number of students. Therefore 'p' is dependent variable.
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Can anyone figure this out? Ive been stuck on it for a while and cant figure out the correct angle
The required scaled copy of polygon B using a scale factor of 0.75 as shown.
What is a scale image?Scale image is defined as a ratio that represents the relationship between the shape and size of a figure and the corresponding dimensions of the actual figure or object.
The polygon B is given as shown, which dimensions are below:
Length of polygon = 8 units
Height of polygon = 10 units
Here, the scale factor = 0.75
So, the dimensions of the scaled copy of polygon B are below:
Length of polygon B' = 8 × 0.75 = 6 units
Height of polygon B' = 10 × 0.75 = 7.5 units
Thus, the scaled copy of polygon B using a scale factor of 0.75 as shown.
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Give your overall description of a box plot
Box plot is a chart that shows data from a five-number summary including one of the measures of central tendency.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
A box plot, also known as a box and whisker plot, is a graphical representation of a set of numerical data that displays its distribution, central tendency, and variability.
The box plot consists of a rectangular box with whiskers that extend from the edges of the box.
The box represents the middle 50% of the data, and the vertical line inside the box represents the median (50th percentile) of the data.
The whiskers extend to the smallest and largest observations that are within 1.5 times the interquartile range (IQR) of the lower and upper quartiles, respectively.
Any observations that fall outside the whiskers are considered outliers and are plotted as individual points.
Box plots are useful for comparing the distribution of data between different groups of data.
Hence, Boxplot is a graphical representation of a set of numerical data that displays its distribution, central tendency, and variability.
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solve for y
2y - 3(2y-3)+2=31
Answer:
y = -5
Step-by-step explanation:
You want to solve for y in 2y -3(2y -3) +2 = 31.
SimplifyParentheses can be eliminated using the distributive property.
2y -6y +9 +2 = 31
Like terms can be combined.
-4y +11 = 31
SolveWe can separate the constant and variable terms by subtracting 11 from both sides.
-4y = 20
The value of y is now found by dividing by -4.
y = 20/(-4) = -5
y = -5
find the lateral surface area of the prism
The lateral surface area of a prism is the sum of the areas of all its rectangular faces.
What is surface area?Surface area is a two-dimensional measure that refers to the total area of a surface, such as the area of a two-dimensional shape, a three-dimensional solid, or a combination of both. It is the sum of the areas of all the faces of a solid object. It is also referred to as the area of the boundary of a three-dimensional object. It can be used to calculate the volume of an object, and is also used in other calculations like area of the base of a triangle, area of a circle, and more.
It is calculated by multiplying the perimeter of the base by the height of the prism. For example, if the base of a prism is a square with side length s and the height is h, then the lateral surface area of the prism can be calculated as 4sh. If the base of the prism is a rectangle with length l and width w, then the lateral surface area of the prism can be calculated as 2lw + 2wh.
The lateral surface area of a prism can also be calculated by adding the areas of the triangular faces. If the base of the prism is a triangle with side lengths a, b, and c, then the lateral surface area of the prism can be calculated as a + b + c.
It is important to note that the lateral surface area of a prism is different from its surface area. The surface area of a prism is the sum of the areas of its lateral faces and its two bases.
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Complete question:
How to find the lateral surface area of the prism?
In triangle ABC, B = 40, a = 7, c = 10. Find the measure of angle C.
We can use the Law of Cosines to solve for angle C:
cos C = (a^2 + b^2 - c^2) / (2ab)
where b is the length of side BC.
We know that B = 40 degrees, so we can find the measure of angle A:
A = 180 - B - C
A = 180 - 40 - C
A = 140 - C
We also know that the sum of the angles in a triangle is 180 degrees, so:
A + B + C = 180
140 - C + 40 + C = 180
180 = 180
This checks out, so we can continue with solving for angle C:
cos C = (a^2 + b^2 - c^2) / (2ab)
cos C = (7^2 + b^2 - 10^2) / (2*7b)
cos C = (49 + b^2 - 100) / (14b)
cos C = (b^2 - 51) / (14b)
We can use the Law of Sines to solve for b:
sin B / b = sin A / a
sin 40 / b = sin (140 - C) / 7
b = sin 40 * 7 / sin (140 - C)
Now we can substitute this value of b into the equation for cos C:
cos C = (b^2 - 51) / (14b)
cos C = [(sin 40 * 7 / sin (140 - C))^2 - 51] / (14 * sin 40 * 7 / sin (140 - C))
Simplifying this equation and solving for cos C, we get:
cos C = 11/14
Therefore, angle C is:
C = cos^-1(11/14)
C ≈ 38.8 degrees
I need this done because i really do not want to fail
Answer:
Step-by-step explanation:
it is 77.6
Please help
At the beginning of spring, Savannah planted a small sunflower in her backyard. When it was first planted, the sunflower was 5 inches tall. The sunflower then began to grow at a rate of 0.5 inches per week. How tall would the sunflower be after 7 weeks? How tall would the sunflower be after w weeks?
Answer:
24.5in
Step-by-step explanation:
7 days in a week. 7 weeks. 7 x 7=49
49 x 0.5 + 24.5in
3. Lisa drives her car 10 km south. She then turns and drives 13 km west. How far is she currently from
her starting place? Sketch a picture, Label the triangle, Set up the problem, Solve.
Using Pythagorean theorem, the distance from where she started is 16.4km
What is Pythagorean TheoremThe Pythagorean Theorem is a mathematical concept that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical terms, the theorem can be expressed as:
c^2 = a^2 + b^2,
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
c² = 10² + 13²
c² = 269
c = √269
c = 16.4 km
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At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 30 km/h. How fast (in km/hr) is the distance between the ships changing at 4:00 p.m.? (Round your answer to three decimal places.)
The distance between the ships is increasing at a rate of approximately 91.6 km/h at 4:00 PM.
How fast is the distance between the ships changing at 4:00 PM?From the question, we have the following parameters that can be used in our computation:
Distance, D = 150 km
Rates = 35 km/h and 30 km/h
Let t be the time elapsed from noon to 4:00 PM
So, we have
t = 4
The distance between the ships to their distances is represented as
d^2 = (D + rate 1 * t)^2 + (rate 2 * t)^2
So, we have
d^2 = (150 + 35t)^2 + (30t)^2
d^2 = (150)^2 + 10500t + 1225t^2 + 900t^2
Differentiate with respect to time (t)
2D d' = 10500 + 2450t + 1800t
So, we have
D d' = 5250 + 1225t + 900t
Substitute 4 for t
D d' = 13750
So, we have
d' = 13750/D
This gives
d' = 13750/150
d' = 91.6
So the distance between the ships is increasing at a rate of approximately 91.6 km/h at 4:00 PM.
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HELP RN PLSPLS!!!! I CANT FIGURE THIS OUTT!!
Answer:m=2/5
Step-by-step explanation:
m[tex]\neq[/tex]0
5/6=1/3m
15m=6
m=2/5, m[tex]\neq[/tex]0
help please. I dont know what a im doing on this problem
The domain of the function is [0, 3) and the range of the function is [-4, 5).
What is Domain and Range of a Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Here the input values in set A is called the domain and the output values in set B is called the range.
Given is a graph of a function.
The input values or domain includes the value of x which forms the curve in the graph.
The output values or range consists of the values of y formed as a result of the input of x.
Look at the x values corresponding to the end points of curve.
The curve extends from x = 0 to x = 3.
But at x = 3, it is an open circle. So x = 3 is not included.
So the input values are from x = 0 to x = 3, where 3 is not included.
Domain in interval notation = [0, 3).
Similarly the values of y extends from y = -4 to y = 5, where 5 is not included, since there is an open circle there.
Range in interval notation = [-4, 5).
Hence the domain and range are [0, 3) and [-4, 5) respectively.
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$300,9%,3 years ??????????????
Answer:
$381
Step-by-step explanation:
9% = 1 year
27% = 3 years
$300 = 100%
After 3 years, we have
100% + 27% = 127%
127% = 1.27
300 tines 1.27 = $381
So, after 3 years has $381
We have this beach bar that has a radius of 20 inches. How much air is inside the ball? 
The air inside the beach ball with a radius of 20 inches is 33493.33 cubic inches.
What is volume of sphere?The capacity of a sphere is its volume. It is the area that the sphere occupies. Cubic measurements of a sphere's volume include m3, cm3, in3, etc. The sphere has a round, three-dimensional shape. Its shape is determined by three axes: the x, y, and z axes. Sports like basketball and football are all examples of spheres with volume.
The volume of the cube is given as:
V = (4/3) πr³
Substituting the value of r = 20 inches we have:
V = (4/3)π(20)³
V = (4/3)(3.14)(20)³
V = 33493.33 cubic inches.
Hence, the air inside the beach ball with a radius of 20 inches is 33493.33 cubic inches.
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A line has a slope of 2 and passes through the point (-4,-4) write its equation in slope intercept form
Answer:
y= 2x+12
Step-by-step explanation:
y=mx+c
y= 2x+12
(0,y) (-4,4)
2=y-4/4
8=y-4
12=y
True or False: The equation x = x2u + x3v, with x2 and x3 free (and neither u or v a multiple of the other), describes a plane through the origin
False. The equation [tex]x = x2u + x3v[/tex] describes a line through the origin, not a plane.
The equation [tex]x = x2u + x3v[/tex] describes a line through the origin, not a plane. This equation can be written as [tex]x = x2u + x3v[/tex], which is the same as [tex]ux2 + vx3 - x = 0[/tex]. This equation is in the form of [tex]ax + by + cz = d[/tex], with a = u, b = v, c = -1, and d = 0. Since the coefficients of [tex]x2, x3[/tex], and x are all non-zero, we can conclude that this equation describes a line, not a plane. To confirm, we can calculate the direction vector for the line. The direction vector is (u, v, -1), which is a single vector and not two or more vectors that would be necessary to describe a plane.
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A snack mix recipe calls for 1 1 2 cups of chips and 1 5 cup of dip. Luke wants to make the same recipe using 1 cup of dip. How many cups of chips will Luke need?
Answer:8
Step-by-step explanation:
15/15=1
112/15=7.466...
you can't buy 7.466... cups, so you round up to 8.
hope I'm right ;)
3. The polynomial function H(x)=x³-4x + 5x-2 has a known linear factor
of (x-1). Use long division and factoring of quadratics to rewrite H(x) as a product of
linear factors.
The polynomial H(x) = x³ - 4x³ + 5x² - 2 divided by (x - 1), will give a quotient of x² - 3x + 2 and a remainder of 0 using the long division, we can write H(x) = (x -1)(x² - 3x + 2).
What is a polynomialA polynomial is a mathematical expression which have a sum of powers in one or more variables with coefficients. The highest power of the variable in a polynomial is called its degree.
We shall divide the polynomial x³ - 4x³ + 5x² - 2 by x - 1 as follows;
x³ divided by x equals x²
x - 1 multiplied by x² equals x³ - x²
subtract x³ - x² from x³ - 4x³ + 5x² - 2 will result to -3x² + 5x - 2
-3x² divided by x equals -3x
x - 1 multiplied by -3x equals -3x² + 3x
subtract -3x² + 3x from x³ - 4x³ + 5x² - 2 will result to 2x - 2
2 divided by x equals 2
x - 1 multiplied by 2 equals 2x - 2
subtract 2x - 2 from 2x - 2 will result to a remainder 0
Therefore, the polynomial H(x) = x³ - 4x³ + 5x² - 2 divided by (x - 1), will give a quotient of x² - 3x + 2 and a remainder of 0 using the long division, we can write H(x) = (x -1)(x² - 3x + 2).
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Find the distance from the point P(2, 1, 4) to the plane through the points Q(1, 0, 0), R(0, 2, 0), and S(0, 0, 3).
The Distance from the point P(2, 1, 4) to the plane is approximately -0.9487.
The equation of a plane can be found using the normal vector and a point on the plane.
First, we can find the normal vector of the plane by finding the cross product of two vectors connecting the points on the plane.
The vector from point Q to R is (0, 2, 0) - (1, 0, 0) = (-1, 2, 0)
The vector from point Q to S is (0, 0, 3) - (1, 0, 0) = (-1, 0, 3)
The normal vector is the cross product of these two vectors:
n = cross(-1, 2, 0), (-1, 0, 3)) = (3, 3, -2)
Next, we can find the equation of the plane using the normal vector and a point on the plane, say Q(1, 0, 0):
ax + by + cz = d
3x + 3y - 2z = d
3x + 3y - 2z = 3(1) + 3(0) - 2(0) = 3
Finally, we can find the distance from the point P(2, 1, 4) to the plane by finding the perpendicular distance between the point and the plane. This can be done using the formula:
[tex]d = (Ax + By + Cz + D)/\sqrt{ {A^2 + B^2 + C^2}[/tex]
[tex]d = (3x + 3y - 2z + 3)/\sqrt{ (3^2 + 3^2 + (-2)^2)[/tex]
[tex]d = (3(2) + 3(1) - 2(4) + 3)/\sqrt{ (3^2 + 3^2 + (-2)^2)[/tex]
[tex]d = (-3)/\sqrt{(18)[/tex]
d = -3/3.162 = -0.9487
So the distance from the point P(2, 1, 4) to the plane is approximately -0.9487.
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The sum of digits of a two-digit number is 9. If the digits are reversed, the number is increased by 63. Find the number.
(a) 18
(b) 27
(C) 36
(d) 72
With the sum of digits of a two-digit number is 9. If the digits are reversed, the number is increased by 63. The number is 18. So, the correct option is A.
Let the two-digit number be represented by 10x + y, where x is the tens digit and y is the ones digit. We are told that x + y = 9 and that if the digits are reversed, the number is increased by 63.
If we reverse the digits of the original number, we get 10y + x. The difference between this number and the original number is 63, so we can set up the equation:
10y + x - (10x + y) = 63
Simplifying this equation, we get:
9y - 9x = 63
Dividing both sides by 9, we get:
y - x = 7
Now we have two equations: x + y = 9 and y - x = 7. We can solve this system of equations by adding the two equations:
2y = 16
y = 8
Substituting y = 8 into x + y = 9, we get:
x + 8 = 9
x = 1
Therefore, the original number is 10x + y = 10(1) + 8 = 18. So the answer is (a) 18.
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