Given:
A graph of a polynomial passes through the point (-1, 60) and has three x-intercepts: (-4,0), (1, 0), and (3, 0).
To find:
The equation of the polynomial.
Solution:
A polynomial is defined as:
[tex]P(x)=a(x-c_1)(x-c_2)...(x-c_n)[/tex]
Where, a is a constant and [tex]c_1,c_2,...,c_n[/tex] are the zeros of the polynomial.
The given polynomial has three x-intercepts (-4,0), (1, 0), and (3, 0). It means [tex]-4,1,3[/tex] are the zeroes of the given polynomial and [tex](x+4),(x-1),(x-3)[/tex] are the factors of the given polynomial.
So, the equation of the polynomial is:
[tex]P(x)=a(x+4)(x-1)(x-3)[/tex] ...(i)
It passes through the point [tex](-1,60)[/tex].
[tex]60=a(-1+4)(-1-1)(-1-3)[/tex]
[tex]60=a(3)(-2)(-4)[/tex]
[tex]60=24a[/tex]
Divide both sides by 24.
[tex]\dfrac{60}{24}=a[/tex]
[tex]2.5=a[/tex]
Putting [tex]a=2.5[/tex] in (i), we get
[tex]P(x)=2.5(x+4)(x-1)(x-3)[/tex]
Therefore, the equation of the given polynomial is [tex]P(x)=2.5(x+4)(x-1)(x-3)[/tex].
How do you find the quotient of a polynomial?
To find the quotient of a polynomial, first divide the polynomial by the divisor. Then use polynomial long division to divide the polynomial by the divisor. The quotient is the result of the polynomial long division.
1. Identify the polynomial to be divided and the divisor.
2. Divide the highest degree term of the polynomial by the divisor.
3. Multiply the divisor by the result from the previous step and subtract this from the polynomial.
4. Bring down the next term in the polynomial.
5. Repeat steps 2-4 until all terms in the polynomial have been divided.
6. The result is the quotient (the answer) of the polynomial division.
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What is the nature of the roots of the discriminant of the equation 4x² 3x 1 0?
The roots of the discriminant of the equation 4x² + 3x + 1 = 0 are complex and imaginary.
The discriminant of a quadratic equation is a value calculated by subtracting the square of the coefficient of the x term from four times the coefficient of the x² term, and then multiplying it by the constant term. For this equation, the discriminant is calculated as D = b2 - 4ac, where b = 3 and c = 1. This gives us D = 32 - 4(4)(1) = -7. Since the discriminant is negative, this means that the roots of the equation are complex and imaginary.
D = b2 - 4ac = 32 - 4(4)(1) = -7
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Find the Slope of (7,-6) and (2,-3)
Answer:
m = -3/5
Step-by-step explanation:
The slope is rise/run or (y2 - y1) / (x2 - x1)
We see the y increase by 3 and the x decrease by 5, so the slope is
-3/5
So, m = -3/5
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Hello! In this question, we are going to find the slope of the line with the coordinates given to us in the question.
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Explanation:
In the question, we were given the points:
(7,-6)(2,-3)In order to find the slope of the line, we will use the slope formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
We will plug our coordinates into the equation and solve.
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Solve:
Plug in the coordinates into the equation and solve.
[tex]m=\frac{-3-(-6)}{2-7} \\\\m=\frac{-3+6}{2-7} \\\\m=\frac{3}{2-7} \\\\m=\frac{3}{-5}\\\\\boxed{m=-\frac{3}{5}}[/tex]
Therefore, our slope for the given coordinates would be [tex]-\frac{3}{5}[/tex]
__________________________________________________________
Answer:
[tex]\boxed{m=-\frac{3}{5}}[/tex]
__________________________________________________________
Additional info:
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What are the roots of the quadratic equation 2x² √ 5x 2 0 Using the quadratic formula?
The roots of the quadratic equation 2x² + 5x - 2 = 0 are -2.7 and 1.3.
The quadratic equation is 2x² + 5x - 2 = 0.
We can solve this equation using the Quadratic Formula:
x = [-b ± Â√(b² - 4ac)] / 2a
Where a = 2, b = 5, and c = -2.
Substituting these values into the formula, we get:
x = [-5 ± Â√(5² - 4(2)(-2))] / 2(2)
x = [-5 ± Â√41] / 4
x = [-5 ± Â√41] / 4
x = [-5 ± 6.4] / 4
x = -2.7 or 1.3
Therefore, the roots of the quadratic equation 2x² + 5x - 2 = 0 are -2.7 and 1.3.
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to use a normal distribution to approximate binomial probabilities, why do we require that both np and n(1 - p) be at least 10?
We require both to ascertain whether the findings are independent, this criterion is applied.
What is Binomial distribution?Statistics frequently employs the binomial distribution as a discrete distribution. When compared to a binomial distribution, the normal distribution is continuous. The chance for 'x' success of an investigation in 'n' trials is represented by the binomial distribution.
As per the given information in the question,
The normal distribution can be used to approximate the binomial distribution. The situation when performing a normal approximation is as follows:
np ≥ 10 and,
np(1 - p) ≥ 10
This criterion is used to determine whether the results are independent.
We are aware that when n and p are combined, the results are independent.
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Two trains going in opposite directions leave at the same time. Train B travels 5 mph faster than train A. In 7 hours the trains are 665 miles apart. Find the speed of each.
Answer:
below
Step-by-step explanation:
x = train A speed
x+5 = train B speed
rate * time = distance
(x + x+5 ) * 7 = 665
14x + 35 = 665
14x = 630
x = 45 mi/hr = A B = 5+ A = 50 mi/hr
Mark’s cost does not vary directly with time spent talking because A graph titled Phone Plan Costs shows Mark's cost and Nikiya's cost and shows Time Spent Talking (hours) on the x-axis and Cost (dollar sign) on the y-axis. A straight line showing Mark's cost starts at (0, 20) and goes through (3, 40). A straight line showing Nikiya's cost starts at (0, 0) and goes through (2, 30).
The phone's cost that varies directly with time are given as follows:
Both Mark and Nikiya.
How to obtain the variation of the functions?The variation of the functions can have two classifications, given as follows:
Direct variation: When the input increases, the output also increases.Inverse variation: When the input increases, the output decreases.Mark's cost starts at (0, 20) and goes through (3, 40), meaning that when the input increases from zero to 3, the output increases from 20 to 40, hence it has a direct variation.
Nikiya's cost starts at (0, 0) and goes through (2, 30), meaning that when the input increases from zero to 2, the output increases from 0 to 30, hence it has a direct variation.
Missing Information
The problem asks for whom's function varies directly with time.
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Answer:
Based on the graph shown to the right, whose phone cost varies directly with the amount of time spent talking?
✔ Nikiya’s only
Mark’s cost does not vary directly with time spent talking because
✔ the cost of 0 hours is not 0 dollars
For Nikiya’s plan, the value of the constant of variation is
15,
which represents Nikiya’s
✔ cost per hour
Step-by-step explanation:
Hope this helps :)
What is the solution to the system of equations y equals 4x 10 y equals two?
The solution of the given system of equation is (3, 2).
In the given question we have to find the solution to the system of equations:
y = 4x − 10 and y = 2
We can solve the system of equation using the substitution method.
In substitution method we have to put the the value of one variable from one equation to the other equation.
We have to equations:
y = 4x − 10........................(1)
y = 2...................................(2)
Now putting the value of y from the equation 2 in equation 1
2 = 4x − 10
Add 10 on both side, we get
4x = 12
Divide by 4 on both side, we get
x = 3
Now putting the value of x in equation 1
y = 4(3) − 10
y = 12 − 10
y = 2
Hence, the solution of the given system of equation is (3, 2).
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The right question is:
What is the solution to the system of equations y = 4x − 10 and y = 2?
What are the common factors of 14 28 and 42?
Which transformation causes the image to be larger or smaller than the pre-image?
A dilation is a type of transformation causes the image to be larger or smaller than the pre-image.
What is dilation?A dilation is a type of transformation that changes the size of a figure without changing its shape. Dilations can make a figure larger or smaller.
In a dilation, the original figure is called the preimage. The dilated figure is called the image. Figures are dilated from a fixed point, called the center of dilation.Dilations also scale all sides of a figure by the same factor. As a result, dilations preserve angle measures and the ratio between corresponding parts.When dilating a figure, the different parts of the figure are multiplied by the scale factor.
In a dilation, the scale factor refers to the ratio between the corresponding sides of the preimage and the image. Therefore, the scale factor, k, can be calculated by dividing the length of one part in the image with the corresponding part in the preimage. The length of the preimage part should always be in the denominator as it belongs to the original figure.
A dilation is a type of transformation causes the image to be larger or smaller than the pre-image.
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If x:(x+3) = 2:3
find x
Answer: -6
Step-by-step explanation:
If x:(x+3) = 2:3, we can set up the following equation:
x / (x+3) = 2/3
Next, we can multiply both sides of the equation by 3 to get rid of the fraction on the right side:
3 * (x / (x+3)) = 3 * (2/3)
This gives us:
x / (x+3) = 2
Finally, we can multiply both sides of the equation by (x+3) to get rid of the fraction on the left side:
x = 2 * (x+3)
We can then solve for x by rearranging the terms on the right side of the equation:
x = 2x + 6
Subtracting 2x from both sides gives us:
x - 2x = 6
This simplifies to:
-x = 6
Finally, we can divide both sides of the equation by -1 to solve for x:
x = -6
Therefore, x = -6.
What is the equation of the line that passes through the point (6, 5) and has a
slope of
-1/3
Answer:
y= -1/3x+7
i graphed it so it should be right
Please help me !!!!!!!!!!!
Answer:
29
Step-by-step explanation:
You have 1,500 Kilometers and 32 15- kilometer races.
Step 1: Divide 1,500 by 15
step 2: Multiply the answer by 10
step 3:
From a window 20 feet above the ground, the angle of elevation to the top of another building is 45°. The distance between the buildings is 52 feet. Find the height of the building to the nearest foot. (no decimal) The building is feet tall.
Using trigonometric ratio, the height of the building is 72 feet
Angle of ElevationThe angle of elevation is an angle that is formed between the horizontal line and the line of sight. If the line of sight is upward from the horizontal line, then the angle formed is an angle of elevation.
In this question, we need to use trigonometric ratio to find the height of the building.
Using tangent of the angle;
tan θ = opposite / adjacent
tan 45 = x / 52
x = 52tan45
x = 52
From the top of the building to the window level of the other building, the distance is 52 feet.
The height of the building = 52 + 20 = 72 feet
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PLS HELP :D here is the question attached
Answer:
y=(2/3) x+7
Step-by-step explanation:
2/3 is the slope which would be parallel
Prove that tan^2A/tan^2A-1+cosec^2A/sec^2A-cosec^2A=1/1-2cos^2A
Prove: <br>
cosec2 A-cos2 A=sec2A -sin2 Atan2A
In the past month Kaylynn rented one video game in for DVDs. The rental price for the video game was $3.30. The rental price for each DVD was $4.30. What is the total amount that Kaitlyn spent on video game and DVD rentals in the past month.
If in the past month Kaylynn rented one video game in for DVDs. the total amount that Kaitlyn spent on video game and DVD rentals in the past month. is : $7.60.
How to find the total amount?We would have to make use of this formula to find the total amount that Kaitlyn spent on video game and DVD rentals in the past month.
Total amount = (Number of video game × Video game rental price ) + Rental price for DVD
Let plug in the formula
Total amount = (1 × $3.30) + $4.30
Total amount = $3.30 + $4.30
Total amount = $7.60
Therefore we can conclude that the total amount is $7.60.
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What is Transform () function used for?
Python's Transform function returns a self-produced data frame with transformed values after applying the function specified in its parameter. This data frame has the same length as the passed data frame.
Now, According to the question:
Python's Transform function returns a self-produced data frame with transformed values after applying the function specified in its parameter. This data frame has the same length as the passed data frame.
What does transform () do?
The transform CSS property lets you rotate, scale, skew, or translate an element. It modifies the coordinate space of the CSS visual formatting model.
The transform() function is used to call function on self producing a Series with transformed values and that has the same axis length as self. Function to use for transforming the data.
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14.
In the figure (not drawn to scale), MO bisects LMN, mZNMO = (6x +19), and m/LMO = (9x -
14). Solve for x and find ZLMN.
A. x= 11, m ZLMN = 198°
B. x= 11, m ZLMN = 170°
C. x=8, m
D. x=8, m
The value of ZLMN is 58.
What is bisects?
Bisection in geometry is the division of an object into two equal or congruent halves, typically by a line, which is subsequently referred to as a bisector. The segment bisector and the angle bisector are the two forms of bisectors that are most frequently used.
LMO+NMO = LMN,
x+34 + NMO = 6x-28
But, MO bisects LMN, so LMO=NMO
x+34 + x+34 = 6x-28
2x+68 = 6x-28
4x = 96
x = 24
That would makem it
24 + 34 =58 T NMO = 6(24)-28
58 + nmo = 144 - 28 = 119
116- 58 = 58
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What is the median of 1 2 3 4 5 6 7?
Answer:
4
Step-by-step explanation:
since there are 7 numbers, divide it by 2. Then you will get 3 remainder 1. Now, count 3 numbers from the first number and the very next number will be your median.
The average test score of RSM students in September increased by 10%. In October it increased by another 15%. By what percent did the average score on the test increase during the two months? PLS PROVIDE SHOW OF STEPS!
The percent that the average score on the test increased during the two months is 12.5%.
By what percent did the average score on the test increase during the two months?It should be noted that mean simply means the average of a set of numbers that are given. In this case, the average test score of RSM students in September increased by 10%. In October it increased by another 15%..
The average of the increase will be:
= (10% + 15%) / 2
= 25% / 2
= 12.5%
The average is 12.5%.
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Is equations 3x 2y 5 and 2x 3y 7 are consistent or inconsistent?
3x + 2y = 5; 2x - 3y = 7. Hence, the given lines are intersecting. So, the given pair of linear equations has exactly one solution and therefore it is consistent.
If a consistent system has an infinite number of solutions, it is dependent . When you graph the equations, both equations represent the same line.
If a system has no solution, it is said to be inconsistent . Represent parallel lines.
if Both line slope are same and parallel to each other.
So, no solution
System is inconsistent.
if Both equation represent the same line. So, infinite many solution.
System is consistent.
if Both lines are parallel. So, no solution.
System is inconsistent.
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By solving simultaneous equations, work out
the coordinates of the point where the two
lines below intersect.
2x+y=8 y
0
y=3x-2
X
The solution of the linear equations 2x + y = 8 and y = 3x - 2 will be (2, 4).
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
The equations are given below.
2x + y = 8 ...1
y = 3x - 2 ...2
From equations 1 and 2, then the value of the variable 'x' is calculated as,
2x + 3x - 2 = 8
5x = 10
x = 2
Then the value of the variable 'y' will be calculated as,
y = 3(2) - 2
y = 6 - 2
y = 4
The solution of the linear equations 2x + y = 8 and y = 3x - 2 will be (2, 4).
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What is the probability that the first roll is an even number and the second roll is a number greater than 4? one-sixth one-third two-thirds five-sixths.
Answer:
2
Step-by-step explanation:
What is the median of the data 46 64 87?
Answer:
64
Step-by-step explanation:
What is the median of the data 46 64 87?
Median: center most value when values are arranged in numerical order so:
46 , 64 , 87
Write the equation of the line in fully simplified slope-intercept form.
Answer:
[tex]y=-3x+7[/tex]
Step-by-step explanation:
[tex]=\frac{4-1}{1-2}=-3 \\ \\ y-4=-3(x-1) \\ \\ y-4=-3x+3 \\ \\ y=3x+7[/tex]
You are making a coffee
table with a glass top surrounded by a cherry border.
The glass is 3 feet by 3 feet. You want the cherry border
to be a uniform width. You have 7 square feet of cherry
to make the border. What should the width of the border be?
Answer:
6 inches
Step-by-step explanation:
You want to know the uniform width of border around a 3 ft square coffee table that would have an area of 7 square feet.
Border areaThe area of the border is x feet wide by 3 feet long along each of the 4 sides, together with 4 corner squares that are x ft square.
A = 7 = 4(3x +x²)
Completing the square, we have ...
x² +3x +9/4 = 7/4 +9/4 . . . . . . divide by 4, add (3/2)² to both sides
(x +3/2)² = 4 . . . . . . . . . . . . simplify
x +3/2 = √4 = 2 . . . . . . square root
x = 1/2 . . . . . . . . . foot
The width of the border should be 6 inches.
Tell weather the p=6 is a solution of p + 5 < 12 yes or no
Answer:
yes
Step-by-step explanation:
Find an equivalent ratio in simplest terms:
45:24
Answer:
15/8
Step-by-step explanation:
The ratio is 45:24
45/24 simplify by 3
15/8
So, the ratio in simplest form is 15/8
If the volume of a soccer ball is 1,930. 75 cubic centimeters, what is the surface area of the soccer ball to the nearest square centimeter?.
The surface area of the soccer ball to the nearest square centimeter is 744.68 square centimeters .
Given :
The volume of a soccer ball is 1,930. 75 cubic centimeters .
we know that ,
shape of soccer ball is same as the sphere
Volume of sphere = 4 / 3 * pi * r^3
1930.75 = 4 / 3 * 3.14 * r^3
1930.75 = 4.18 * r^3
r^3 = 1930.75 / 4.18
= 462
r = 7.7 cm
Surface area of soccer ball = 4 * pi * r^2
= 4 * 3.14 * 7.7^2
= 12.56 * 59.29
= 744.68 square centimeters
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