The partial fraction expansion of (x³ + 4)/ ((x² - 1)(x² + 3x + 2)) is -3/ 4(x + 1) + 5/12(x - 1) + 4/3(x + 2) - 3/ 2(x + 1)². So the coefficients A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2.
A partial fraction expansion is a way of expressing a rational function (a function that can be written as the ratio of two polynomials) as the sum of simpler fractions, each with a numerator that is a constant or a simple polynomial. The process of finding the partial fraction expansion of a rational function is also known as partial fraction decomposition.
The denominator is
(x² - 1)(x² + 3x + 2) = (x² - 1)(x² + x + 2x + 2)
(x² - 1)(x² + 3x + 2) = (x + 1)(x - 1)(x + 1)(x + 2)
So by partial fraction expansion
(x³ + 4)/ ((x² - 1)(x² + 3x + 2)) = (x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2))
(x³ + 4)/ ((x + 1)(x - 1)(x + 1)(x + 2)) = A/ (x + 1) + B/(x - 1) + C/(x + 2) + D/ (x + 1)²
A(x - 1)(x + 1)(x + 2) + B(x + 1)²(x + 2) + C(x + 1)²(x - 1) + D(x - 1)(x + 2) = x³ + 4
(A + B + C)x³ + (2A + 4B + C + D)x² + (-A + 5B - C + D)x + (-2A + 2B - C - 2D) = x³ + 4
Thus from coefficients of x³, x², x and 1,
A + B + C = 1
2A + 4B + C + D = 0
-A + 5B - C + D = 0
-2A + 2B - C - 2D = 4
Solving
A = -3/4, B = 5/ 12, C = 4/3 and D = -3/2
--The question is not clear, the question is given below--
"Enter (A,B,C,D) in order below if A, B, C, and D are the coefficients of the partial fractions expansion of
[tex]$$12\cdot\frac{x^3 + 4}{(x^2-1)(x^2 +3x+ 2)}[/tex] "
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fill in the blank
The vertext of y= x^2-8x+18 is ( , )
PleASE HELP ASAP
Answer:
Vertex is at (4, 2)
Step-by-step explanation:
to find the 'x' value of the vertex, use 'x = -b/2a'
in this problem, a = 1, b = -8, c = 18
x = -(-8)/2(1)
x = 8/2
x = 4
plug 2 in for 'x' in equation and solve for 'y':
y = 4² - 8(4) + 18
y = 16 - 32 + 18
y = -16 + 18
y = 2
Vertex is at (4, 2)
Determine the intercepts of the line. Do not round your answers. y-4=7(x-6)y−4=7(x−6)y, minus, 4, equals, 7, left parenthesis, x, minus, 6, right parenthesis xxx-intercept: \Big((left parenthesis ,,comma \Big))right parenthesis yyy-intercept: \Big((left parenthesis ,,comma \Big))
The y-intercept of the line y = 7x -38 is -38. And the x-intercept is 0
What is a slope?In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.
Given:
The equation of the line,
y-4=7(x-6)
Simplify,
y -4 = 7x - 42
y = 7x -38
And slope-intercept form:
y = mx + b,
where m is the slope and y is the intercept.
Comparing with the slope intercept form,
we get,
m = 7 and b = -38
And the x-intercept is 0
Therefore3, y-intercept is -38.
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Mr. Barth is painting an arrow on the school parking lot. He draws the edges between the following points on the coordinate plane: (–2, 2), (5, 2), (5, 6), (12, 0), (5, –6), (5, –2), (–2, –2).
Answer:
sorry
Step-by-step explanation:
PLEASE HELP PICTURE ATTACHED!!
IF m /3 = (9x + 14) and m/5 = (14x - 1), find m/28.
Answer:
∠8 = 139°
Step-by-step explanation:
Before we start solving, let's identify the figure to understand the problem.
We are given two parallel lines that are cut by a transversal.
Two lines are parallel if they have the same arrow head marking.
If two parallel lines are cut by a transversal, the following angles are created:
alternate interior anglesalternate exterior anglesvertical anglesadjacent anglescorresponding anglesI have attached an image of the different types of angles to this response.
Thus, we can use the types of angles to solve for ∠8.
∠3 and ∠5 are corresponding angles. Therefore, they have the same measure.
Set up the equation to solve for x:
∠3 = ∠5
9x + 14 = 14x - 1 . . . . . . substitute the values
14 = 5x - 1 . . . . . . . . subtract 9x from the LHS and RHS
15 = 5x . . . . . . . . . . add 1 to the LHS and RHS
∴ 3 = x . . . . . . . . . . . . divide the LHS and RHS by 5 to isolate x
Now that we know what x is, we can substitute it into the value of ∠3.
∠3 = 9x + 14
∠3 = 9(3) + 14 . . . . substitute the values
∠3 = 27 + 14 . . . . . compute
∴ ∠3 = 41
∠3 and ∠8 are adjacent angles. Therefore, they add up to 180°.
Set up the equation to solve for ∠8:
∠8 + ∠3 = 180
∠8 + 41 = 180 . . . . substitute the known values
∴ ∠8 = 139° . . . . . subtract 41 from the LHS and RHS to isolate ∠8
⭐if this response helped you, please mark it the "brainliest"!⭐
Answer:
Step-by-step explanation:
180-54=139 so 8 equals 139
A ladder is leaning against a wall. The top of the ladder is 999 feet (\text{ft})(ft)(, start text, f, t, end text, )above the ground. If the bottom of the ladder is moved 3\,\text{ft}3ft3, start text, f, t, end text farther from the wall, the ladder will be lying flat on the ground, still touching the wall. How long, in feet, is the ladder
999.06ft999.06 in feet is the ladder.
What is Pythagorean Theorem?Pythagorean Theorem is a mathematical formula used to calculate the length of the sides of a right triangle. It states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side, which is known as the hypotenuse. The formula is written as a² + b² = c², where a and b are the two shorter sides, and c is the hypotenuse.
To find the length of the ladder, use the Pythagorean Theorem. The length of the ladder is equal to the square root of the sum of the squares of the side lengths of the right triangle formed by the ladder and the wall.
The two side lengths of the right triangle are 999\,\text{ft}999ft999, start text, f, t, end text and 3\,\text{ft}3ft3, start text, f, t, end text.
Therefore, the length of the ladder is equal to the square root of (999^2+3^2), which is equal to 999.06\,\text{ft}999.06ft999.06,
start text, f, t, end text.
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I really also need help on this question lol
Answer:
13/15
For this question, you could type it into the calculator.
Hence, this question is too trivial, and your question might get deleted as a result. Since this is your first question, I will help you out but subsequently, do use a calculator to solve this kind of questions.
Step-by-step explanation:
4 1/3 can be converted to 13/3 as an improper fraction.
Next, evaluate 13/3 divided by 5
(13/3) / 5 = (13/3) x (1/5)
= 13/15
Answer:
Step-by-step explanation:
There are a number of ways to do this. I am going to take a little longer, but I am hoping that it will make sense to you.
4[tex]\frac{1}{3}[/tex] ÷ 5
Another name for 1 is [tex]\frac{3}{3}[/tex] I am going to rewrite 4[tex]\frac{1}{3}[/tex] knowing this.
[tex]\frac{3}{3}[/tex] + [tex]\frac{3}{3}[/tex] + [tex]\frac{3}{3}[/tex] + [tex]\frac{3}{3}[/tex] + [tex]\frac{1}{3}[/tex] = [tex]\frac{13}{3}[/tex]
Another way to write 5 is [tex]\frac{5}{1}[/tex].
I want both of my fraction to have the same denominator so I will multiple
[tex]\frac{5}{1}[/tex] x [tex]\frac{3}{3}[/tex] = [tex]\frac{15}{3}[/tex] I am now ready to divide by fractions
[tex]\frac{13}{3}[/tex] ÷ [tex]\frac{15}{3}[/tex] Divide across
[tex]\frac{\frac{13}{15} }{\frac{3}{3} }[/tex] = [tex]\frac{\frac{13}{15} }{1}[/tex] = [tex]\frac{13}{15}[/tex]
If x = 9 units, y = 12 units, and z = 7 units, then what is the volume of the right rectangular pyramid shown above?
Answer:
"252 units³" is the appropriate solution.
Step-by-step explanation:
The given values are:
Length,
x = 9 units
width,
y = 12 units
height,
z = 7 units
As we know,
The volume of rectangular pyramid will be:
⇒ [tex]Volume=\frac{1}{3}(length\times width\times height)[/tex]
On substituting the values, we get
⇒ [tex]=\frac{1}{3}\times 9\times 12\times 7[/tex]
⇒ [tex]=3\times 85[/tex]
⇒ [tex]=252 \ units^3[/tex]
Carlos has completed 18 deliveries so far this week. He needs to make 60 deliveries for the week. What percentage of his deliveries has Carlos completed?
Answer:
30%
Step-by-step explanation:
18/60 = 0.30
0.30 = 30%
Let C be between D and E. Use the Segment Addition Postulate to solve for n.
DC = 5n – 30
CE = 6n – 25
DE = 44
A. N = 13
B. N = 4
C. N = –5
D. N = 9
Segment Addition Postulate states that n has a value of 9 when DC = 5n - 30 and CE = 6n - 25 and DE = 44.
what is equation ?A mathematical equation is a formula that connects two statements using the equal sign (=) to denote equivalence. An algebraic equation is a mathematical statement that establishes the equality of two mathematical expressions. For example, the variables 3x + 5 and 14 are separated by an equal sign in the equation 3x + 5 = 14. Mathematical equations are used to express the connection between two phrases on either side of a letter. There is frequently just one variable, which is also the symbol. example: 2x - 4 = 2.
given
DE = DC + CE
44 = 5n – 30 + 6n – 25
44 = 11n - 55
44 + 55 = 11n
n = 9
Segment Addition Postulate states that n has a value of 9 when DC = 5n - 30 and CE = 6n - 25 and DE = 44.
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question: which month has an average low temperature that is cooler than December’s? Explain your answer.
thanks to anyone who helps! giving out brainliest as well :)
Answer:January
Step-by-step explanation: if it is asking to explain based on data given, you can show that -11.6<-10.4, but if it is asking based on other information that you recieved, i cant really help other then to tell you to look at the cylces of day and night, because some parts of alaska get no sun for 6 months, so that could be why january is averagly colder
tossing a coin and spinning a spinner with five equal sections, 1-5
Answer:
Its gonna be a 50/50 on the coin and 20/100 chance for the spinner
Answer:
5 outcomes for the spinner and a 50/50 chance for the coin
Step-by-step explanation:
for the spinner, there are five numbers, 1,2,3,4, and 5. so, there is a 1/5 chance or you could also say there is a 20/100 chance. For the coin, there are two sides and 100 divided by 2 is 50 therefore, there is a 50/50 chance
Hope this helps!!
3 cm and 9 mm converted completely into centimetres to one decimal place is ____cm. Someone please help me it would be greatly appreciated :)
Chee and her children went into a grocery store and she bought $12 worth of bananas and mangos. Each banana costs $0.50 and each mango costs $2. She bought twice as many bananas as mangos. Graphically solve a system of equations in order to determine the number of bananas, x,x, and the number of mangos, y,y, that Chee bought.
We get the number of bananas as 8 and the number of mangoes as 4.
What is the general equation of a straight line?The general equation of a straight line is -
y = mx + c
{m} - slope.
{c} - intercept along the y - axis.
Given is that Chee and her children went into a grocery store and she bought $12 worth of bananas and mangos. Each banana costs $0.50 and each mango costs $2. She bought twice as many bananas as mangos.
We can write the system of equations as -
x = 2y ..... (1)
(1/2)x + 2y = 12 ..... (2)
We can write equation (2) as -
(1/2) x 2y + 2y = 12
3y = 12
y = 4
then -
x = 8
Therefore, we get the number of bananas as 8 and the number of mangoes as 4.
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Set builder notation integers between 7 and 50
Answer:
[tex]\{x|x\in \mathbb{Z}, 7 < x < 50\}[/tex]
Step-by-step explanation:
Basically, the above answer tells us that the set of all x such that x belongs to Z, the set of integers, and x is between 7 and 50.
6. Lines C and D below are cut by transversal, t. Which of
the following is true about the angles formed?
Answer:
m∠8 = m∠2, option G
Step-by-step explanation:
Angles 8 and 2 are alternate interior angles, which means they are equal. Thus, the answer is option G, m∠8 = m∠2
This composite figure is made up of three simpler shapes. What is the area of the figure? 68 square cm 83 square cm 59 square cm 33 square cm
Answer:
68 square cm
Step-by-step explanation:
the composite is made up of a triangle , rectangle and parallelogram
area of composite = area of triangle + area of rectangle + area of parallelogram
Area of parallelogram = base x height
9 x 2 = 18
area of triangle = 1/ 2 x base x height
1/2 x 6 x 5 = 15
Area of a rectangle = length x width
5 x 7 = 35
15 + 35 + 18 = 68
Two students wrote the equations shown below.
Erica Aliyah
y = -0.5x y = 2.5x - 8
Which statement is true?
A Graphs of both equations will pass through the origin.
B Only Erica’s equation is proportional.
C Both equations have a negative slope.
D Only Aliyah's is proportional
Only Erica's equation (y = -0.5x) is proportional
What is an equation?An equation is used to show the relationship between numbers and variables.
The standard form of a linear equation graph is:
y = mx + b
Where m is the slope of the line and b is the y intercept.
Given the Erica equation:
y = -0.5x
The slope is -0.5, hence it is a negative slope and the intercept is at 0, hence it passes through the origin and is proportional
Given the Aliyah equation:
y = 2.5x - 8
The slope is 2.5, hence it is a positive slope and the intercept is at -8, hence it does not pass through the origin and is not proportional
Both equations are linear equations
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What is the value of cos x
Answer fastttttt
Answer:
cos x = 30/34 or 15/17
Step-by-step explanation:
SOH CAH TOA
side a is 30 (adjacent)
side h is 34 (hypotenuse)
cos= a/h
A bucket contains 4 red marbles, 10 blue marbles, 7 yellow marbles, and 5 green marbles.
What is the probability that you will reach in and pick a red marble?
a 4/25
b 4/23
c .2563
d 15.3%
Answer:
d. 15.3%
Step-by-step explanation:
There are a total of 4 + 10 + 7 + 5 = 26 marbles
P(pick a red) = 4/26 = 2/13 = .153 = 15.3%
Augustus has made a pile of 2,000 candies, and he is giving away 25 candies a day from this pile. When will he have 350 candies left in this pile
Mashika: We already know from polling data that some segments of the electorate provide significant support to Ms. Puerta. If those segments also provide significant support to Mr. Quintana, then no segment of the electorate that provides significant support to Mr. Quintana provides significant support to Mr. Ramirez.
Salim: But actually, as the latest polling data conclusively shows, at least one segment of the electorate does provide significant support to both Mr. Quintana and Mr. Ramirez.
Among the following statements, which is it most reasonable to infer from the assertions by Mashika and Salim?
A. At least one segment of the electorate provides significant support neither to Mr. Quintana nor to Mr. Ramirez.
B. At least one segment of the electorate provides significant support to Ms. Puerta but not to Mr. Quintana.
C. Each segment of the electorate provides significant support to Ms. Puerta.
D. Each segment of the electorate provides significant support to Mr. Quintana.
E. Each segment of the electorate provides significant support to Mr. Ramirez.
Answer:
D. Each segment of the electorate provides significant support to Mr Quintana
Step-by-step explanation:
From the question, we have the following logic statement;
If those segment that provide significant support to Ms Puerto also provide support to Mr. Quintana then no segment of the electorate that provides significant support to Mr. Quintana provides significant support to Mr. Ramirez
Counter to the If Then Logic structure
At least one segment of the lectorate provides significant support to Mr. Quintana and Mr. Ramirez
Therefore, there is at least one segment that provides support to Mr. Quintana but not to Ms. Puerta
The segments that provide support to Ms. Puerta do not provide support to Mr. Ramirez
Mr. Quintana's support includes those from the supporters of Ms. Puerta and those from the supporters of Mr. Ramirez
Therefore, each segment of the electorate provide significant support to Mr. Quintana.
Shaheen ha many triangular piece, each of area 1 q. Unit and many rectangular piece, each of area 2 q. Unit
If Shaheen has many triangular piece, each of area 1 sq. Unit and many rectangular piece, each of area 2 sq. Unit she arranges these to form the given shape then the area of shape is 14 Square Units .
The Area of one triangular piece is given as = 1 Sq . Unit ;
The Area of one rectangular piece is = 2 Sq. Unit ;
From the shape given below ,
we can see that the shape has 4 triangular pieces and 5 rectangular pieces .
So , total area of given shape is ⇒[tex]4\times Area Of Triangle + 5\times Area Of Rectangle ;\\[/tex]
Substituting the values of Area ,
we get ;
[tex]Total Area = 4\times 1 +5\times 2[/tex]
= [tex]4+10[/tex]
= 14 Sq . Units .
Therefore , the area of the given shape is 14 Sq. Units .
The given question is incomplete , the complete question is
Shaheen has many triangular piece, each of area 1 sq. Unit and many rectangular piece, each of area 2 sq. Unit .
If she arranges these to form a shape as shown below, what would be the area of this shape ?
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Help me please I got 10 minutes
The triangle LJK is congruent to the triangle RTS.
What do congruent triangles mean?
Triangles that are congruent share the same matching side lengths and angle measurements. The conditions for triangle congruence are: SSS, SAS (Side, Side, Side) (Side-Angle-Side)
According to the given image of triangles,
We can observe that:
∠L=∠R, ∠T=∠J, ∠S=∠K.
We can also observe that TS=JK, LK=RS, RT=LJ.
Therefore according to SSS congruency ΔLJK ≅ ΔRTS.
According to SAS congruency ΔLJK ≅ ΔRTS.
According to ASA congruency ΔLJK ≅ ΔRTS.
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What is 3 and 2/3 as a mixed number?
3 and 2/3 as a mixed number will be 11/3.
Define integers.Zero, a positive natural number, or a negative integer denoted by a minus sign are all examples of integers. The inverse additives of the equivalent positive numbers are the negative numbers. The set of integers is frequently represented in mathematical notation by the boldface Z or blackboard bold Z.
What is a mixed number?A mixed number is a combination of a whole number and a fraction. It represents a number that is greater than zero but less than the next whole number. A mixed number is written with the whole number followed by a space and then the fraction.
[tex]=3+2/3[/tex]
[tex]=(3*3+2)/3[/tex]
[tex]=(9+2)/3[/tex]
[tex]=11/3[/tex]
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Use the FOIL Method to find (b+3)(b+7).
Step-by-step explanation:
968866j5x4cctyvcyvyvycycc
There was a population of 40 bears in a National Park. After a year, the population increased by 40%. How many bears are there in the park now?
Answer: 56
Step-by-step explanation:
Given
There are 40 bears in a national park
After a year, Population increased by 40%
An increase in population is
[tex]=40\times 40\%\\=40\times 0.4\\=16[/tex]
So, a new population is the sum of the original and increased population
[tex]\Rightarrow 40+16\\\Rightarrow 56[/tex]
Calculate the size of the angle marked 'x' in the pentagon shown below. Give your answer in degrees (°).
Answer: x= 96 degrees
Step-by-step explanation:
It we figure out the rest of the angles they would be (the outside ans inside angles are straight angles):
123+153+49+119= 444
Since the total of angles in a pentagon = 540 degrees
540-444 96 degrees
The measure of an angle x will be equal to 96°.
What is the Pentagon?A Pentagon is a two-dimensional geometric shape that consists of five straight sides and five angles. It is a type of polygon, which is a closed shape with three or more sides.
The Pentagon is often used to represent the shape of buildings, monuments, and symbols in architecture and design.
If we figure out the rest of the angles they would be (the outside and inside angles are straight angles):
123+153+49+119= 444
Since the total of angles in a pentagon = 540 degrees
540-444= 96°
Therefore, the measure of an angle x will be equal to 96°.
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Brittany practices hitting softballs for 2/3 of an hour each day for three days.For how many hours does she practice hitting softballs?
Step-by-step explanation:
2/3 hours every day for 3 days.
so, for the total hours we need to multiply 2/3 by 3 (3 times 2/3 hours) :
3 × 2/3 = 2 hours
you can get there directly by saying that the factor of 3 and the denominator of 3 are canceling each other out. what is left is 2 in the numerator.
or you can do the full calculation and say
3 × 2/3 = 3×2/3 = 6/3 = 2
Kathy wants to place a printer from the filing cabinet. Can the other two distances shown be and ? Identify the answer with the correct explanation.
No. the sum of any two lengths of the triangle is greater than the third length.
Option (A) is correct.
What is a triangle?A triangle is a polygon with three edges and three vertices. In Euclidean geometry, a triangle is completely determined by the three points that make up its vertices.
Kathy wants to place a printer 9ft from the filing cabinet. Can the other two distances are shown to be 7ft and 2ft.
No the sum of any two lengths of the triangle should be greater than the third side.
but 7 + 2 = 9, so the condition for a triangle is false.
Hence, the correct explanation would be "No. the sum of any two lengths of the triangle is greater than the third length."
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Kathy wants to place a printer 9ft from the filing cabinet. Can the other two distances shown be 7ft and 2ft? Identify the answer with the correct explanation. Printer Desk Filing Cabinet
O Yes; the sum of any two lengths is greater than the third length.
O No; the sum of any two lengths is not greater than the third length.
O Yes; the sum of any two lengths is equal to the third length.
O No; the sum of any two lengths is greater than the third length
Answer:
No; The sum of any two lengths is NOT greater than the third length
Step-by-step explanation:
It is given that three objects, a desk, a filing cabinet, and a printer, form a triangle.
Use the Triangle Inequality Theorem to determine whether the given distances can be the lengths of the sides of a triangle.
9+7=16>2
9+2=11>7
7+2=9=9
The sum of one pair of lengths is equal to the third length. Therefore, the distances between the desk, the printer, and the desk and the filing cabinet cannot be 7ft and 2ft since 7+2 equals 9.
a paint roller has a width of 12 inches and a radius of 3 inches, what is the surface area that can be painted with one complete rotation of the roller
The surface area that can be painted with one complete rotation of the roller is 226.19 inches².
Surface area is defined as the total amount of area that covers the surface or outside of a three-dimensional figure.
A paint roller is in the shape of a cylinder. To determine the surface area that can be painted with one complete rotation of the roller, solve for the surface area of a cylinder without the circular bases.
SA = 2πrh
where SA = surface area
r = radius of the base = 3 inches
h = height of the cylinder = width = 12 inches
Plug in the values and solve for the surface area.
SA = 2πrh
SA = 2π(3 inches)(12 inches)
SA = 226.19 inches²
Hence, the surface area that can be painted with one complete rotation of the roller is 226.19 inches².
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