The operation in the equation, (r + s)(r2 - rs + s2), is solved by distributing the multiplication of the binomial across each term in the trinomial. When the multiplications are complete and the terms are added, the resulting solution is r3 + s3.
Explanation:The indicated operation in the question is the multiplication of the binomial, (r + s), and the trinomial, (r2 - rs + s2). This can be performed by using the distributed property of multiplication over addition. In this case, you would multiply each term in the binomial by each term in the trinomial, then add the products. So the solution will be:
r3 - r2s + rs2 + r2s - rs2 + s3 = r3 + s3
Note: the terms -r2s and r2s cancel each other out, as do the terms rs2 and -rs2, which is why the final result is r3 + s3.
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greatest common factor of −27x2yz5 + 15x3z3
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The data set below represents the total number of touchdowns a quarterback threw each season for 10 seasons of play.
29, 5, 26, 20, 23, 18, 17, 21, 28, 20
1. Order the values:
5, 17, 18, 20, 20, 21, 23, 26, 28, 29
2. Determine the median:
= = 20.5
Calculate the measures of variability for the data set.
The range is ___ touchdowns. The interquartile range is ____ touchdowns.
The range is 24 and the interquartile range is the difference between the median of the second-half to the first-half is 8.
What is a median?The median of the data is the middle value of the data which is also known as the central tendency of the data and is known as the median.
The data set below represents the total number of touchdowns a quarterback threw each season for 10 seasons of play.
29, 5, 26, 20, 23, 18, 17, 21, 28, 20
Arrange in ascending order, we have
5, 17, 18, 20, 20, 21, 23, 26, 28, 29
The range will be given as
→ Range = 29 - 5 = 24
The interquartile range will be given as
→ Interquartile range = median of second-half - median of first-half
→ Interquartile range = 26 - 18
→ Interquartile range = 8
The range is 24 and the interquartile range is 8.
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the value 3 is an upper bound for the zeros of the function shown below. f(x)=-3x^3+20x^2-36x+16 True or Flase
The statement "The value 3 is an upper bound for the zeros of the function f(x) = -3x^3 + 20x^2 - 36x + 16" is FALSE.
To determine whether the value 3 is an upper bound for the zeros of the function f(x) = -3x^3 + 20x^2 - 36x + 16, we need to check if the function has any real roots greater than 3.
One way to approach this is by analyzing the behavior of the function as x approaches infinity. We can check the sign of the leading coefficient (-3) and the constant term (16) to determine the overall behavior of the function.
Leading coefficient:
The leading coefficient of -3 indicates that the highest power of x in the function is negative. This means that as x approaches infinity, the function will decrease without bound.
Constant term:
The constant term of 16 indicates that the function intersects the y-axis at y = 16.
Considering these observations, we can infer that the function starts at a positive value (y = 16) and approaches negative infinity as x increases. This implies that the function f(x) = -3x^3 + 20x^2 - 36x + 16 will have at least one real root greater than 3.
Therefore, the statement "The value 3 is an upper bound for the zeros of the function f(x) = -3x^3 + 20x^2 - 36x + 16" is FALSE.
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A surveyor measures the lengths of the sides of a triangular plot of land. What is the measure of the angle of the triangular plot at which the surveyor stands? Approximate to the nearest degree.
Answer: B
Step-by-step explanation:
What is the x intercept for y=3x+4
13. What is a simpler form of each product?
(4x - 6y^3)^2
(A). 16x^2 - 24xy^3 + 36y^6
(B). 16x^2 - 48xy^3 + 36y^6
(C). 16x^2 + 36y^6
(D). 16x^2 - 4xy^3 + 36y^6
14. The area of a rectangular garden is given by the trinomial x^2 + 6x - 27. What are the possible dimensions of the rectangle? Use factoring.
(A). x - 9 and x + 3
(B). x + 9 and x - 3
(C). x - 9 and x - 3
(D). x + 9 and x + 3
15. The area of a rectangular garden is given by the trinomial x^2 + x - 30. What are the possible dimensions of the rectangle? Use factoring.
(A). (x - 6) and (x - 5)
(B). (x + 6) and (x - 5)
(C). (x + 6) and (x + 5)
(D). (x - 6) and (x + 5)
What is the factored form of the following expressions?
16. x^2 - 10xy + 24y^2
(A). (x + 6y)(x + 4y)
(B). (x - 2y)(x + 12y)
(C). (x + 2y)(x - 12y)
(D). (x - 6y)(x - 4y)
17. The area of a rectangular barnyard is given by the trinomial 6x^2 + 7x - 20. What are the possible dimensions of the barnyard? Use factoring.
(A). 2x - 5 and 3x + 4
(B). -2x + 5 and -3x + 4
(C). 2x + 5 and 3x - 4
(D). 2x - 5 and 3x - 4
18. The area of a rectangular carpet is given by the trinomial 5x^2 - 3x - 14. What are the possible dimensions of the carpet? Use factoring.
(A). (5x + 7) and (-x - 2)
(B). (5x + 7) and (x - 2)
(C). (5x - 7) and (x - 2)
(D). (5x - 7) and (x + 2)
The answer to question 13 is (B). [tex]\(16x^2 - 48xy^3 + 36y^6\).[/tex]
The answer to question 14 is (A). [tex]\(x - 9\) and \(x + 3\)[/tex]
The answer to question 15 is (A). [tex]\((x - 6)\) and \((x - 5)\).[/tex]
The answer to question 16 is (A). [tex]\((x + 6y)(x + 4y)\).[/tex]
The answer to question 17 is (A).[tex]\(2x - 5\) and \(3x + 4\).[/tex]
The answer to question 18 is (B). [tex]\((5x + 7)\) and \((x - 2)\).[/tex]
To find a simpler form of the product [tex]\((4x - 6y^3)^2\)[/tex], we apply the formula for squaring a binomial, which is [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. Here, [tex]\(a = 4x\)[/tex] and [tex]\(b = 6y^3\).[/tex]
So, [tex]\((4x - 6y^3)^2 = (4x)^2 - 2(4x)(6y^3) + (6y^3)^2\).[/tex]
Calculating each term, we get:
[tex]\((4x)^2 = 16x^2\),[/tex]
[tex]\(-2(4x)(6y^3) = -48xy^3\),[/tex]
[tex]\((6y^3)^2 = 36y^6\).[/tex]
Putting it all together, we have:
[tex]\(16x^2 - 48xy^3 + 36y^6\).[/tex]
To find the possible dimensions of the rectangle, we need to factor the trinomial [tex]\(x^2 + 6x - 27\).[/tex] We look for two numbers that multiply to -27 and add up to 6. These numbers are 9 and -3.
So, [tex]\(x^2 + 6x - 27 = (x + 9)(x - 3)\).[/tex]
We factor the trinomial [tex]\(x^2 + x - 30\)[/tex] by finding two numbers that multiply to -30 and add up to 1. These numbers are 6 and -5.
So, [tex]\(x^2 + x - 30 = (x - 6)(x + 5)\).[/tex]
To factor [tex]\(x^2 - 10xy + 24y^2\)[/tex], we look for two numbers that multiply to \[tex](24y^2\)[/tex] and add up to -10y. These numbers are -6y and -4y.
So, [tex]\(x^2 - 10xy + 24y^2 = (x - 6y)(x - 4y)\)[/tex].
To find the possible dimensions of the barnyard, we factor the trinomial [tex]\(6x^2 + 7x - 20\).[/tex] We need two numbers that multiply to [tex]\(6 \times -20 = -120\)[/tex] and add up to 7. These numbers are 15 and -8. We then split the middle term accordingly and factor by grouping:
[tex]\(6x^2 + 15x - 8x - 20 = 0\),[/tex]
[tex]\(3x(2x + 5) - 4(2x + 5) = 0\),[/tex]
[tex]\((3x - 4)(2x + 5)\).[/tex]
We factor the trinomial [tex]\(5x^2 - 3x - 14\)[/tex] by finding two numbers that multiply to [tex]\(5 \times -14 = -70\)[/tex] and add up to -3. These numbers are -10 and 7. We then split the middle term accordingly and factor by grouping:
[tex]\(5x^2 - 10x + 7x - 14 = 0\),[/tex]
[tex]\(5x(x - 2) + 7(x - 2) = 0\),[/tex]
[tex]\((5x + 7)(x - 2)\).[/tex]
What is the value of the function, then determine if the graph opening up or down f(x)=-5(x+7)^2+6?
A. a= -5, opens down
B. a= -5, opens up
C. a= 7, opens up
D. a= 7, opens down
Factor the expression. 21x^2 + 43x + 20
(7x – 5)(3x – 4)
(7x + 5)(3x + 4)
(7x – 5)(3x + 4)
(7x + 5)(3x – 4)
Roberto's toy car travels at 40 centimeters per second (cm/sec) at high speed and 25 cm/sec at low speed. If the car travels for 30 seconds at high speed and then 51 seconds at low speed, what distance would the car have traveled?
Answer:
2475 cm
Step-by-step explanation:
We are given that Roberto's toy car travels 40 cm/sec at high speed and 25 cm/sec at low speed.
We have to find that the distance would have the car traveled
Speed of car at high speed=40 cm/sec
Speed of car at low speed=25 cm/sec
If car takes time to travel at high speed=30 seconds
If car takes time to travel at low speed=51 seconds
[tex]Distance=speed\times time[/tex]
Using this formula
Distance traveled by the car at high speed=[tex]40\times 30=1200 cm[/tex]
Distance traveled by the car at low speed=[tex]51\times 25=1275 cm[/tex]
Total distance traveled by the car =1200+1275=2475 cm
Hence, the distance would have traveled by the car=2475 cm
If somebody uses 1 quart of blue paint each month in one year how many gallons of paint will they use
Dwayne's garden is triangle-shaped with two equal sides and a third side that is 4 ft more than the length of an equal side. If the perimeter is 49 ft, how long is the longest side?
The value of Ari's rolls of coins is $113.00. If pennies and dimes come in rolls of 50 coins each, and nickels and quarters come in rolls of 40 coins each, which of these combinations could Ari have?
A. 5 rolls of pennies, 8 rolls of nickels, 4 rolls of dimes, and 7 rolls of quarters
B. 4 rolls of pennies, 8 rolls of nickels, 7 rolls of dimes, and 5 rolls of quarters
C. 4 rolls of pennies, 8 rolls of nickels, 5 rolls of dimes, and 7 rolls of quarters
D. 5 rolls of pennies, 8 rolls of nickels, 7 rolls of dimes, and 4 rolls of quarters
Answer:
Option C is correct.
Step-by-step explanation:
Given is :
The value of Ari's rolls of coins is = $113
The coins are pennies, dimes, nickels and quarters.
Total money is represented by = penny + nickle + dime + quarter All values in dollars are represented by:
113 = .01* pennies + .05* nickles + 0.1* dimes + 0.25* quarters
Further calculating we get,
113 = .01* 50*penny rolls + .05 * 40*nickle rolls + .1 * 50*dime rolls + .25 * 40*quarter rolls
[tex]113=.5p+2n+5d+10q[/tex]
where p is the number of penny rolls, n is the number of nickle rolls, d is the number of dime rolls, and q is the number of quarter rolls
Now checking all the options by putting values.
A. [tex]113=.5(5)+2(8)+5(4)+10(7)[/tex]
[tex]113\neq 108.5[/tex]
B. [tex]113=.5(4)+2(8)+5(7)+10(5)[/tex]
[tex]113\neq 103[/tex]
C. [tex]113=.5(4)+2(8)+5(5)+10(7)[/tex]
[tex]113=113[/tex]
D. [tex]113=.5(5)+2(8)+5(7)+10(4)[/tex]
[tex]113\neq 93.5[/tex]
Therefore, option C is the right option.
A swing set is going to be placed over a region of mulch that is shaped like a trapezoid. The bases of the trapezoid have a length of 12 and 15 feet, and the perpendicular distance between the bases is 8.5 feet. What is the area of the region under the swing set? If necessary, round your answer to the nearest tenth.
Answer:
[tex]114.8\ ft^{2}[/tex]
Step-by-step explanation:
we know that
The area of a trapezoid is equal to
[tex]A=\frac{1}{2}(b1+b2)h[/tex]
In this problem we have
[tex]b1=12\ ft[/tex]
[tex]b2=15\ ft[/tex]
[tex]h=8.5\ ft[/tex] ----> the height of the trapezoid is the perpendicular distance between the bases
substitute the values
[tex]A=\frac{1}{2}(12+15)(8.5)[/tex]
[tex]A=\frac{1}{2}(27)(8.5)=114.75\ ft^{2}[/tex]
Round to the nearest tenth
[tex]114.75=114.8\ ft^{2}[/tex]
Answer:
114.8 is the answer
Step-by-step explanation:
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a system of two equations contains one quadratic equation and one linear equation. the quadratic system of the equation is y=x^2+5x-9. the solutions of the system are (3,15) and (-1,-13). what is the linear equation in the system?
Help if you don't mind!
rodney is given two linear equations : x - y = 11 and 2x +y =19. what value if x shkuld he get as a solution for thus system of linwar equations
Answer: 10
Step-by-step explanation:
Match each economic term with its description. Tiles There are no barriers to entry in the market. There is a single seller in the market. Three companies secretly enter into a price agreement. Every company in this market structure is aware of the actions of the other companies. Pairs monopoly arrowBoth oligopoly arrowBoth perfect competition arrowBoth collusion arrowBoth \
Each economic term is matched with its appropriate description as follows:
OLIGOPOLY || There are no barriers to entry in the market.
MONOPOLY || There is a single seller in the market.
COLLUSION || Three companies secretly enter into a price agreement.
PERFECT COMPETITION || Every company in this market structure is aware of the actions of the other companies.
There are no barriers to entry in the market is the match of - Oligopoly (The market is shared by many sellers )
There is a single seller in the market is the match of - Monopoly
Three companies secretly enter into a price agreement is the match for - Collusion
Every company in this market structure is aware of the actions of the other companies is the match for Perfect Competition.
Larry has 62 nickels, 24 dimes, 17 quarters, and 11 fifty-cent pieces. How much money does he have?
A. $17.15
B. $16.30
C. $15.25
D. $16.75
Answer:
Larry has 1525 cents or 15.25 dollars.
Step-by-step explanation:
The following are the values for each of the coins:
1 Nickel=5 cents
1 Dime = 10 cents
1 Quarter = 25 cents
1 Fifity-cent = 50 cents
So the total value of the Larry's have is
Total money = 62*5 cents + 24*10 cents + 17*25 cents + 11*50 cents
Total money = 310 cents + 240 cents + 425 cents + 550 cents
Total money = 1525 cents
This is equal to 15.25 dollars.
Use technology or a z-score table to answer the question.
The expression P(z<1.45) represents the area under the standard normal curve below the given value of z.
What is P(z<1.45)?
Answer:
P(z < 1.45) ≈ 0.92647
Step-by-step explanation:
Several forms of technology are available for finding the area under the standard normal curve. There are probability apps, web sites, spreadsheets, and calculator functions.
Technology requirementsThe area under the standard normal curve between two values of z is given on many spreadsheets and by many calculators using the normalcdf(a,b) function. In this form, 'a' is the lower bound, and 'b' is the upper bound of the z-values for which the area is wanted.
For the problem at hand, the value of 'a' is intended to be negative infinity. A calculator allows input of no such value, so some "equivalent" value must be used. (At least one calculator manual suggests -1e99.)
The area of the normal curve below z=-8 is less than 10^-11, so -8 is a suitable stand-in for -∞ on a calculator that displays a 10-decimal-digit result. All the decimal digits shown are accurate, not affected by our choice of lower bound.
Calculator value of P(z < 1.45)The attachment shows the value of the expression is about ...
P(z < 1.45) ≈ 0.92647
Answer with explanation:
We have to find , P (z< 1.45).
Breaking ,z value into two parts, that is , In the column,the value at, 1.40 and in the row ,value at , 0.05,the point where these two value coincide,gives value of Z<1.45.
The value lies in the right of mean.
So, P(z<1.45)=0.9265
In the,Normal curve, at the mid point of the curve
Mean =Median =Mode
Z value at Mean = 0.5000
→So, if you consider , the whole curve,
P(Z<1.45)= 0.9265 × 100=92.65%=92%(approx) because we don't have to consider ,z=1.45.
→But, if you consider, the curve from mean ,that is from mid of the normal curve
P (z<1.45)=92.65% - 50 %
=42.65% =42 %(approx) because we don't have to consider ,z=1.45.
The function f(x) = 8(1/4)^x is reflected across the y-axis to create g(x). Which table of values could be used to graph g(x)?
Answer:
Table 1
Step-by-step explanation:
We have the function [tex]f(x)=8(\frac{1}{4})^{x}[/tex].
Now, the function g(x) is obtained by reflecting f(x) across y-axis.
i.e. g(x) = f(-x)
i.e. [tex]g(x)=8(\frac{1}{4})^{-x}[/tex]
So, substituting the values of x in f(x) or g(x), we will discard some options.
2. For x=0, the value of [tex]f(0)=8(\frac{1}{4})^{0}[/tex] i.e. f(0) = 8.
As in table 2, f(0) = 0 is given, this is not correct.
3. For x=0, the value of [tex]g(0)=8(\frac{1}{4})^{-0}[/tex] i.e. g(0) = 8.
As in table 3, g(0) = -8 is given, this is not correct.
4. For x=0, the value of [tex]g(0)=8(\frac{1}{4})^{-0}[/tex] i.e. g(0) = 8.
As in table 3, g(0) = 0 is given, this is not correct.
Thus, all the tables 2, 3 and 4 do not represent these functions.
Hence, table 1 represents f(x) and g(x) as the values are satisfied in this table.
Math help With please
The value of y directly varies with x, and y=5.4 when x =9. Find y when x= negative 10
Final answer:
The value of y, which directly varies with x, is found by first determining the constant of variation when x = 9 and y = 5.4. Using this constant, we calculate the value of y for x = -10, resulting in y = -6.
Explanation:
The value of y directly varies with x, which means the relationship between x and y can be described by the equation y = kx, where k is the constant of variation. Since y = 5.4 when x = 9, we first find the constant of variation as follows: k = y/x = 5.4/9 = 0.6. Now, to find y when x is -10, we use the constant of variation k in the equation: y = kx = 0.6(-10) = -6.
The relationship between y and x is one of direct variation, represented by the equation y = kx, where k is the constant of variation. Given that y = 5.4 when x = 9, the constant k is calculated as 0.6. Applying this constant, when x = -10, the value of y is found to be -6. This process showcases the direct variation principle in determining y based on the given x values and the constant of variation.
The equation of line A is y = 7x + 12. Line B is perpendicular to line A and passes through the point . What would be the solution to the system of equations represented by line A and line B?
The solution to the system of equations involving line A (y = 7x + 12) and a perpendicular line B passing through a given point involves finding the negative reciprocal of the slope of line A for line B, using the point-slope formula to get line B's equation, and solving the system to find their point of intersection.
Explanation:The question revolves around finding the equation of line B, which is perpendicular to line A (y = 7x + 12) and passing through a given point, then solving the system of equations formed by lines A and B. First, we identify that the slope of line A is 7. Since line B is perpendicular to line A, its slope will be the negative reciprocal of 7, which is -1/7. Assuming the point it passes through is provided in the question, we can use the point-slope formula y - y1 = m(x - x1) to find the equation of line B. After determining the equation of line B, we solve the system of equations represented by lines A and B to find the point of intersection, which will be the solution to the system.
To solve the system of equations, we would set the equations equal to each other and solve for one variable, then substitute back to find the other variable. This process involves algebraic manipulation such as substitution or elimination method. The final solution will be the coordinates (x,y) where both lines intersect.
Polygon ABCD has vertices A(0, 2), B(0, 8), C(7, 8), and D(7, 2). What is polygon ABCD and its perimeter? (Hint: Draw the polygon on the coordinate plane and find the slopes of each side.)
A. rectangle; P = 26 linear units
B. square; P = 42 units2
C. parallelogram; P = 42 linear units
D. trapezoid; P = 26 linear units,
Answer:
Option A.
Step-by-step explanation:
The vertices of Polygon ABCD are A(0, 2), B(0, 8), C(7, 8), and D(7, 2).
Plot all vertices on the coordinate place.
If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the slope of the line is
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
First we need find the slope of each sides, using the above formula.
[tex]m_{AB}=\frac{8-2}{0-0}=\infty[/tex]
[tex]m_{BC}=\frac{8-8}{7-0}=0[/tex]
[tex]m_{CD}=\frac{8-2}{7-7}=\infty[/tex]
[tex]m_{AD}=\frac{2-2}{7-0}=0[/tex]
The slope of vertical lines is ∞ and slope of horizontal line is 0. It means sides AB and CD are vertical lines and sides BC and AD are horizontal lines.
Vertical and horizontal lines are perpendicular to each other. It means all interior angles of the polygon are right angles.
From the below figure it is clear that
[tex]AB=6[/tex]
[tex]BC=7[/tex]
[tex]CD=6[/tex]
[tex]AD=7[/tex]
Opposite sides are equal and interior angles are right angles, so the polygon is a rectangle.
Perimeter of the polygon is
[tex]Perimeter=AB+BC+CD+AD[/tex]
[tex]Perimeter=6+7+6+7=26[/tex]
Perimeter of polygon ABCD is 26 linear units.
Therefore, the correct option is A.
Please help! Thanks!
What is the similarity ratio of a cube with volume 1,728m3 to a cube with volume 19,683m3?
A. 9:4
B. 4.9
C. 144:729
D. 729:144
A grid shows the positions of a subway stop and your house. The subway stop is located at (6,-2) and your house is located at (3,1) what is the distance to the nearest unit between your house and the subway stop?
A. 10
B. 9
C. 4
D. 3
Simplify 3 ∙ 2x. What is the coefficient?
plz help
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Celia earned $5.00. She saved $1.00 and spent the rest. What is the ratio of the amount saved to the amount spent?
Solution :
Given that, Celia earned $5.00.
She saved $1.00 and spent the rest.
To find the ratio of the amount saved to the amount spent , we must first calculate the amount spent.
To calculate the amount spent, subtract the amount saved from the total money earned.
Amount spent by Celia = amount earned - amount saved [tex] = 5-1 = 4 [/tex]
[tex] ratio= \frac{amount\:saved}{ amount\:spent} =\frac{1}{4} \\
\\
ratio= 1:4 [/tex]
Hence, 1:4 is the ratio of the amount saved to the amount spent.
miles is buying a chair that regularly costs $250. today the chair is on sale for 30% off. if the tax rate is 6%, what is the sale price of the chair including tax?
The sale price of the chair including tax is $185.5.
What is tax?In mathematics, the tax calculation is related to the selling price and income of taxpayers. It is a charge imposed by the government on the citizens for the collection of funds for public welfare and expenditure activities. There are two types of taxes: direct tax and indirect tax.
Given that, Miles is buying a chair that regularly costs $250.
Today the chair is on sale for 30% off
So, the new cost is 250-30% of 250
= 250 - 30/100 ×250
= 250-75
= $175
The tax rate is 6%
Sale price = 175+6% of 175
= 175+6/100 ×175
= 175+0.06×175
= $185.5
Therefore, the sale price of the chair including tax is $185.5.
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