The number of real roots found are-
1. (x-4)(x+3)^2(x-1)^3=0 ; 6 real roots.
2. X^2(x^3-1)=0 ; 3 real roots.
3. X(x+3)(x-6)^2=0 ; 4 real roots.
4. 3x(x^3-1)^2=0 ; 3 real roots.
5. (x^3-8)(x^4+1)=0 ; 1 real roots.
Define the term roots of polynomial?If a root is not imaginary (defined as a number as in form a + bi, whereby I is a number which is not on any real number line, which is why the name imaginary), it is said to be real.Consider the given cases-
1. (x-4)(x+3)^2(x-1)^3=0,
Put each equals zero.
x – 4 = 0 or (x + 3)^2 = 0 or (x – 1)^3 = 0,
Thus,
x = 4
x = –3 twice
x = 1 thrice
All are real numbers, thus given polynomial has 6 real roots.
2. x^2(x^3-1)=0
Put each equals zero.
x² = 0
x³ – 1 = x³ – 1³ = (x – 1)(x² + x + 1) = 0
The quadratic expressions has 2 imaginary roots.
Use the discriminant formula to find-
D = b² – 4ac = 1² – 4(1)(1) = 1 – 4 = –3 < 0.
Thus, roots that are not real.
So, the real roots of this polynomial : 0(twice) and 1.
Hence, thus given polynomial has 3 real roots.
3. x(x+3)(x-6)^2=0
Put each equals zero.
x = 0
x = 3
x = 6 twice
Hence, thus given polynomial has 4 real roots.
4. 3x(x^3-1)^2=0
Put each equals zero.
3x = 0 and x = 0
(x^3 – 1)² = [(x – 1)(x² + x + 1)]² = (x – 1)²(x² + x + 1)² = 0,
Here, only two roots are real
The quadratic expression do have unreal roots, thus 1 twice are the real roots.
Hence, thus given polynomial has 3 real roots..
5. (x^3-8)(x^4+1)=0
Put each equals zero.
x³ – 8 = x³ – 2³ = (x – 2)(x² + 2x + 4) = 0
There is only one real root i.e, 2, the quadratic expression do have imaginary roots
Check using the discriminant formula
D = b² – 4ac = 2² – 4(1)(4) = 4 – 16 = – 12 < 0.
x⁴ + 1 = 0 has imaginary roots.
Hence, thus given polynomial has 1 real roots..
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The cost of a pizza varies directly as the square of its radius. If a
pizza with a 6 inch radius costs $8.00, how much should a pizza with
an 11 inch radius cost?
The cost of a pizza with an 11 inch radius is $26.86
In this question, the cost of a pizza varies directly as the square of its radius.
Let C be the cost of pizza and r be the radius of Pizza.
We have been given the direct proportionality between the cost of pizza and the radius of Pizza.
C ∝ r²
If a pizza with a 6 inch radius costs $8.00
8 = k * 6^2
where k is the proportionality constant.
8 = k * 36
k = 0.222
We need to find the cost of pizza with an 11 inch radius.
so, we get an equation,
C = k * 11^2
substitute above value of k
C = 0.222 * 11^2
C = 0.222 * 121
C = $26.86
Therefore, the cost of a pizza with an 11 inch radius is $26.86
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A
If m ZABD = 75° and m/ABC = 40°,
then mZCBD = [?]°
We see BC is in the middle of the lines AB and BD, therefore mABC + mCBD = mABD.
We're given mABD = 75, mABC = 40.
Therefore, mCBD = mABD - mABC = 75 - 40 = 35.
Answer:
∠CBD = 35°
Step-by-step explanation:
We are told in the question that m∠ABD = 75° and m∠ABC = 40°. We are then asked what the value of m∠CBD is.
As we can see from the diagram, ∠ABD is made up of the angles ABC and CBD. Therefore:
∠ABD = ∠ABC + ∠CBD
⇒ 75° = 40° + ∠CBD
⇒ 75° - 40° = 40° + ∠CBD - 40° [
⇒ 35° = ∠CBD
∴ ∠CBD = 35°
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>
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In the following diagram, BC is parallel to DE.
What is the measure of Zx?
Angles are not necessarily drawn to scale.
B
D
F
54°
105°
A
E
And please explain how I get the answer pls
The value of Angle DAF, x is 21°
Given,
Line BC parallel to line DE
The value of Angle ABC = 54°
The value of Angle BAC = 105°
We have to find the value of angle DAF
Here,
Angle DAF = x
Now,
Angle ABC = Angle DAB (Alternate interior angles)
So,
Angle DAB = 54°
Then,
Angle DAF, x = 180 - (105 + 54) (Linear pair of angles)
Linear Pair of Angles;
When two lines cross at one point, a linear pair of angles is created. If the angles follow the intersection of the two lines in a straight line, they are said to be linear. A linear pair's total angles are always equal to 180°.
That is,
Angle DAF, x = 180 - 159
Angle DAF, x = 21°
Therefore,
The value of Angle DAF, x is 21°
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the letters that spell mississippi are each written on separate tiles lying facedown on a table. a tile is selected at random, the letter is recorded, and then the tile is placed facedown on the table. then the process repeats. what is the theoretical probability of choosing a tile with the letter p? responses 14 1 fourth 411 4 over 11 111 1 over 11 211
The theoretical probability of choosing a tile with the letter p is 1/8
The letter of the word MISSISSIPPI are written on separate tiles lying down on the table and tile is chosen at random and the the letter is noted and then the tile is lied again on the table.
Probability of the event is,
P(E) = Total favorable outcomes/Total possible outcomes.
Total possible outcomes in this case are 11 because there are a total of 11 letters.
The total favorable outcomes that the chosen letter is P is,
Total favorable outcomes = ¹¹C₁/(2!)(2!)(2!)
Total favorable outcomes = 11/8
Probability P that chosen number is P is,
P = (11/8)/11
P = 1/8
So, the probability that the chosen tile is with letter P is 1/8.
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marjore needs to earn a C in her class. her current test scores are 67,78,75,78 and .her final exam is worth 6 test scores. In order to earn a C marjorties average must lie between 70 and 79 inclusive. What range of scores can marjorie's receive on the final exam to earn a C in the course?
The range of scores that Marjorie's can receive in the final exam to earn a C in the course is 67 to 82 .
in the question ,
it is given that
the Marjorie's score in the tests are 67 , 78 , 75 , 78 .
let the score in the final exam be "x"
it is given that the final exam score is worth 6 test scores , which means
the score in final exam is = 6x
and the total number of tests = 6+4 = 10
given that to get the grade C minimum average is = 70
that is (67 + 78 + 75 + 78 + 6x)/10 = 70
67 + 78 + 75 + 78 + 6x = 700
298 + 6x = 700
6x = 700 - 298
6x = 402
x = 402/6
x = 67
given that to get the grade C maximum average is = 79
that is (67 + 78 + 75 + 78 + 6x)/10 = 79
67 + 78 + 75 + 78 + 6x = 790
298 + 6x = 790
6x = 790 - 298
6x = 492
x = 492/6
x = 82
the range of scores is 67 to 82 .
Therefore , The range of scores that Marjorie's receive in the final exam to earn a C in the course is 67 to 82 .
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A sphere has a diameter of 24 mm.
What is the volume of the sphere?
48pi mm³
192pi mm³
2304pi mm³
I don't know.
Answer:
Step-by-step explanation:
volume of a sphere = ⁴/3 π r³. r = D/2. V = 4/3 × π × (D/2)³. V = 4/3 × π × D³/8. V = ⅓ × π × D³/2. V = ⅓× π × (24mm)³/2.
What is the interest on a $1900 investment at 3% for 1 year?
The interest is $57 on a $1900 investment at 3% for 1 year
Given,
In the question:
Principal = $1900
Rate of interest = 3%
Time = 1
Now, According to the question:
Principal = $1900
Rate of interest = 3%
Time = 1
Substitute:
Principal = $1900
Rate of interest = 3%
Time = 1 into formula Interest = P x R x T
I = 1900 x 0.03 x 1
Calculate the product or quotient:
I = 57
Hence, The interest is $57 on a $1900 investment at 3% for 1 year.
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PR is a chord of a circle Center 0 and radius 20CM - POR = 120° calculate (a) length of the Chord PR (b) length of are PQR
Answer:
i have the similar question just radius is changed frm 12 to 20 cm..
hopefully this will help u ...^_^
17 ÷ 2578
Please help I need an answerrrr
Answer: 17/2578 (Decimal: 0.006594)
Step-by-step explanation:
That's the answer :)
Which statement is true?
This table does not show a probability distribution.
This table shows a probability distribution.
After using the probability distribution table, the given table does not show a probability distribution. The statement 1 is correct.
In the given question, we have to check which statement is correct regarding to the given option.
The given statements are:
(1) This table does not show a probability distribution.
(2) This table shows a probability distribution.
Before checking the statement we learn about a probability distribution table.
In a probability distribution table every result of a statistical experiment is connected to the likelihood that the result will actually occur. An experiment's result is recorded as a random variable, which is typically written with a capital letter.
As we can see that there is given the probability of having the blue, green, yellow and red color.
So the given table does not show a probability distribution.
So the statement 1 is correct.
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how do i solve 1) −10+ r = 3+r-7
Answer: No Solution
Explanation: 10 + r = 3 + r - 7
Combine like terms: 10 + r = -4 + r
Subtract R to both sides: 10 = 4
There is NO SOLUTION
Help please
Find the values of the variables and the measures of the angles.
Answer:
X= 29 because it is a right angle, and all right angles are = to 90, so you subtract 61 from 90.Y= 29Z= 61
Step-by-step explanation:
Answer:
Step-by-step explanation:
Summation of all angles within a triangle is equal to 180 degrees.
y+61+90=180
Therefore, y = 180-61-90 = 29 degrees
61+x = 90
Therefore, x = 90-61 = 29 degrees
z+90+29 = 180
Therefore, z = 180 - 90 - 29 = 61 degrees
A housepainter mixed 5 gal of blue paint with every 9 gal of yellow paint in order to make a green paint. Which ratio of gallons of blue paint to gallons of yellow paint will NOT make the same shade of green paint?
The ratio of gallons of blue paint to gallons of yellow paint will NOT make the same shade of green paint 4:9.
What is Ratio?A ratio is a comparison between two amounts that is calculated by dividing one amount by the other. The quotient a/b is referred to as the ratio between a and b if a and b are two values of the same kind and with the same units, such that b is not equal to 0. Ratios are represented by the colon symbol (:). As a result, the ratio a/b has no units and is represented by the notation a: b.
Given:
A housepainter mixed 5 gal of blue paint with every 9 gal of yellow paint.
So, the ratio of gallons of blue paint to gallons of yellow paint is
= 5 to 9 or 5:9
Now, ratio of gallons of blue paint to gallons of yellow paint will NOT make the same shade of green paint
= 1- 5/9
=4/9
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8. If a population doubles every 5 years, how many years will it take for the population to increase by 10 times
its original amount?
First: If the population gets multiplied by 2 every 5 years, what does it get multiplied by each year? Use this
to help you answer the question.
After ≈16.61 years the population will increased by times it's original amount.
What do you mean by population?
A population is the entire set of people in a group, whether that group is a country or a collection of people who share a certain trait. A population is the group of people from which a statistical sample is taken in statistics.
Given: If a population doubles every 5 years.
Assume, 'A' is the starting population, and 't' is the passing time in years.
Given the fact that a population doubles every five years, our population is growing exponentially, as per the question.
Here, the equation can be written as follows: [tex]A(t) = A(2)^{\frac{t}{5} }[/tex] (1)
We now need to determine when the population will have multiplied by 10 by that time.
Let's substitute [tex]A(t) = 10A[/tex] in the above equation.
[tex]10A = A(2)^{\frac{t}{5} }[/tex]
[tex]10 = (2)^{\frac{t}{5} }[/tex]
Taking the natural log of both sides we get:
[tex]ln10 = \frac{t}{5} \times ln2[/tex] {[tex]ln(a)^n= nln(a)[/tex]}
2.3025 x 5 = t x 0.6931
[tex]t = \frac{2.3025}{0.6931}\\t = 16.6096[/tex]
Therefore, After ≈16.61 years the population will increased by times it's original amount.
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help meeeeeeeeeeeeeeeeeee pleaseeee rn rnnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
help meeeeeeeeeeeeeeeeeee pleaseeee rn rnnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
how many 44-digit positive integers (that is, integers between 10001000 and 99999999, inclusive) having only even digits are divisible by 55?how many 44-digit positive integers (that is, integers between 10001000 and 99999999, inclusive) having only even digits are divisible by 55?
There are 100 4-digit positive integers are possible between 1000 and 9999.
Our first step is to divide the four digit number into four places: tens, hundreds, and thousands.
To solve this problem, we must start from the right-hand side and find how many even numbers should be placed where the desired answer can be obtained.
There are a total of [0, 2, 4, 6, and 8] objects
There should only be one digit on one place, since it is the only digit that makes the number divisible by five
There are 1 ways to fill the one's place, so the total number is 1.
It is possible to take any number from the given sets of even numbers in the tens place,
There are five ways to fill one place, so there are five ways to fill it.
Taking a look at the hundreds place,
All numbers can be placed in the hundreds place from the given sets, so all numbers can be taken from the hundreds place.
Therefore, there are five ways to fill the hundreds place.
Taking a look at the thousands,
It is possible to fill thousands of places with 4 numbers from a given set of even numbers.
Therefore, there are four ways to fill the thousands place
The next step is to use the concept of factorial to find the number of ways to make a 4-digit number from even numbers that are divisible by 5.
To fill the respective places, we will multiply all the ways we calculated
= 4 × 5 × 5 × 1
= 100
Hence, there 100 4-digit positive integers possible.
Question
how many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5?
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In the figure below, a pole has two wires attached to it, one on each side, forming two right triangles.
Based on the given information, answer the questions below. Round each answer to the nearest tenth.
Part A: How tall is the pole? Show your work.
Part B: How far from the base of the pole does Wire 2 attach to the ground? Show your work.
Part C: How long is Wire 1? Show your work.
A. Applying the trigonometric ratio, the length of pole = 29.6 ft.
B. 23.1 ft.
C. Length of wire 1 = 45.1 feet.
How to Apply the Trigonometric Ratios?The trigonometric ratios are:
sin ∅ = opposite length/hypotenuse length cos ∅ = adjacent length/hypotenuse lengthtan ∅ = opposite length/adjacent length.Part A: Apply the trigonometric ratio, tan ∅ = opp/adj, to find the length of the pole
∅ = 41°
Opp. = length of pole
Adj. = 34 ft
Substitute
tan 41 = length of pole/34
Length of pole = (tan 41)(34) = 29.6 ft.
Part B:
Let the distance between the base of the pole and wire 2 be represented as x. Therefore, apply tan ∅ = opp/adj.
∅ = 38
Opp. = x
Adj. = 29.6 ft
Therefore:
tan 38 = x/29.6
x = (tan 38)(29.6)
x = 23.1 ft.
Part C: Use the Pythagorean theorem to find the length of wire 1.
Length of wire 1 = √34² + 29.6²)
= 45.1 feet.
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in a study, physicians were asked what the odds of breast cancer would be in a woman who was initially thought to have a 1% risk of cancer but who ended up with a positive mammogram result (a mammogram accurately classifies about 80% of cancerous tumors and 90% of benign tumors.) what is the probability of a woman having a malignant (cancerous) tumor in her breast if she has a positive mammogram result? round to 3 decimal places as needed.
The probability that a woman has a malignant tumor in her breast is 7.5%. ie 75
Let + = mammogram result is positive
B = tumor is benign
M = tumor is malignant.
Note that Bc = M.
given that P(M) = .01,
so P(B) = 1 − P(M) = 0.99.
We are also given the conditional probabilities
P(+ | M) = .80 and
P(− | B) = .90, where the event − is the complement of +
P(+ | B) = .10
Bayes’ formula:-
P(M | +) = P(+ | M) P(M) / (P(+ | M)P(M) + P(+ | B)P(B))
= 0.80 × 0.01 / (0.80 × 0.01 + 0.10 × 0.99)
=0.075
So the chance would be 7.5%. ie 75
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A roofing contractor uses shingles at a rate of 3 bundles for every 96 square feet of roof covered. At this rate, how many bundles of shingles will he need in order to cover a roof that is 416 square feet?
Answer:
13
Step-by-step explanation:
416/96 = 4.33333
416 is 4.333 times 96, so he needs 4.333 times 3 bundles.
4.3333 × 3 = 13
Answer: 13
Answer:
13
Step-by-step explanation:
you see how much it covers initially. (96/3=32) then 416/32=13
2 times 221 divide 4 ?
if x is a number in the interval 2 6 state all integers that satisfy the given inequality
Answer:
it is because I haven't completed the answer from THE WORK OF THE BEST ways to solve a questionStep-by-step explanation:
do I
PLEASE HELP! WILL GIVE BRAINLEST! IM BEGGIN YOUUUU. It’s just creative writing not english work!
Teresa estimates that the height of a giraffe in a zoo is 18 feet. In fact, it is 16 feet tall. What is Teresa’s percentage error? Use proportional relationships to solve the problem. Show all your work and round your answer to the nearest tenth.
Answer: 11.1%
Step-by-step explanation:
The formula for percent error is:
|Actual-predicted/predicted| * 100%
In the context of the problem:
Actual = 16 ft
Predicted = 18 ft
Plug these values into the equation:
|16-18/18| * 100% = percent error
11.11% = percent error
If you have any other problems about percent error, you should be able to just plug these values into the formula
Which is equivalent to 3√8x?
It is equivalent to 8^x/3
What is equivalent equations?OBJECTIVES FOR EQUALITY EQUATIONSAfter finishing this section, you need to be able to:Sort an equation into a conditional or an identity category.Mentally solve simple equations.Check to see if some equations are equal.A mathematical equation is a statement that two numerical expressions are equal.Equations go into one of two categories.acc to our question-
to write a square root into fraction. To do that, you just have to apply the formula :
x^power/root
Then, here it gives . 8^x/3
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Which relation is a function?
The relation that is a function in the option is graph C.
What is a function?A function relates an input to an output.
A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.
In a simpler term, a function relates each element of a set with exactly one element of another set (possibly the same set).
In a function, the x value is exactly related to a single y value, exactly, means that a function is single valued. It will not give back 2 or more results for the same input.
Therefore, the only graph that follows the rule of a function is graph C.
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What is another way to write x^(x^2)? How do I work with this? I don't understand
The other ways of rewriting the expression are (xˣ)⁽²⁾ or xˣ²
How to rewrite the expression?From the question, we have the following expression that can be used in our computation:
x^(x^2)
Rewrite the expression properly
So, we have the following representation:
x⁽ˣ⁾⁽²⁾
The only ways the above expression can be rewriting the expression is by changing the positions of the brackets or removing the brackets entirely
Other than that, there are no other ways of rewriting the expression
So, we have
(xˣ)⁽²⁾ or xˣ²
Hence, the expressions are (xˣ)⁽²⁾ or xˣ²
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Which of the following rational functions is graphed below?
Answer:
C
Step-by-step explanation:
In the graph, you can see an asymptote is happening for x=-5, so look for an equation that divides by 0 for x=-5. That is clearly C, for x=-5 the denominator is 0.
Find the slope of the line that passes through the pair of points (8, −2), (1, 1)
The slope of the line that passes through the given pair of points is, -3/7.
What is slope of the line?
A line's slope in mathematics is defined as the ratio of the change in the y coordinate to the change in the x coordinate. Both the net change in the y-coordinate and the net change in the x-coordinate are denoted by y and x, respectively. where "m" represents a line's slope. So, the slope of a line is tanθ.
Given: (8, -2), (1, 1)
Since, if two points [tex](x_1, y_1), (x_2, y_2)[/tex] are given then slope (m) is defined as,
[tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex].
Here,
[tex](x_1, y_1) = (8, -2)\\(x_2, y_2)=(1, 1)[/tex]
So, the slope can be defined as,
[tex]m = \frac{1-(-2)}{1-8} \\m=\frac{3}{-7} \\m=-\frac{3}{7}[/tex]
Hence, the slope of the line that passes through the given pair of points is, -3/7.
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y≥x²+2x-8 factor the quadratic equation to find x intercepts
The factors of the quadratic equation are x≥-4, x≥2 and lies in the interval (-∞, -4]∪[2, ∞)
What is meant by quadratic equation?A quadratic equation is any equation that can be rearranged in standard form as ax²+bx+c=0 where x represents an unknown value and a, b, and c represent known numbers. In general, a = 0; equations with a = 0 are considered degenerate because the equation becomes linear or even simpler. The numbers a, b, and c are the equation's coefficients, and they can be distinguished by referring to them as the quadratic coefficient, linear coefficient, and constant or free term, respectively.
To find x-intercept y=0
y≥x²+2x-8
x²+2x-8≥0
x²+4x-2x-8≥0
x(x+4)-2(x+4)≥0
(x-2)(x+4)≥0
∴x≥-4, x≥2
∴(-∞, -4]∪[2, ∞)
Therefore, the factors of the quadratic equation are x≥-4, x≥2 and lies in the interval (-∞, -4]∪[2, ∞)
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The triangle on the right is a scaled copy of the triangle on the left. Identify the scale factor. Express your answer as a fraction in simplest form.
The scale factor in the simplest form of a fraction of the triangles will be 43/78.
What is dilation?Dilation is the process of increasing the size of an item without affecting its form. Depending on the scale factor, the object's size can be raised or lowered.
There is no effect of dilation on the angle.
The triangle on the right is a scaled duplicate of the triangle on the left.
The ratio of the side of a big triangle to the side of a small triangle. Then we have
Scale factor = 43 / 78
The scale factor in the simplest form of a fraction of the triangles will be 43/78.
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