Given -
f(x) = x⁵ - 4x³ + 7x² + 3x - 5
To Find -
How many positive real zeroes does f(x) have =?
Step-by-Step Explanation -
We have
f(x) = x⁵ - 4x³ + 7x² + 3x - 5
where the signs of coefficients are:
+ - + + -
We can see there are three changes in the sign in f(x).
So, from
Descarte's rule,
there are either 3 0r 1 positive real roots
Now,
f(-x) = -x⁵ + 4x³ + 7x² - 3x - 5
Signs: - + + - -
So, here f(-x) has two sign changes.
From this we can conclude there is at least 1 real root and either 4, 2 or 0 imaginary roots.
Final Answer -
Positive real zeroes f(x) have =
Minimum = 1
Maximum = 3
HELP QUICK
For each statement about owners of equity in a business, select True or False.
00000 Replace the side length of this square with 4 in., and find the area. S
The area of the square is 16 square inches.
The area of a square is calculated by multiplying a side by itself.
In this case, the side S is replaced by 4 in, then:
[tex]\begin{gathered} \begin{cases}A=S\cdot S \\ S=4in\end{cases} \\ A=4in\cdot4in=16in^2 \end{gathered}[/tex]20 students started the class. Then 2 students dropped the
class.
What percent of the students have dropped the class?
What quadrant is 0 And. . -1 1/2 in or is it on a y- axis or x-axis.
The coordinate pair is between the third and fourth quadrants, on the y-axis.
In which quadrant is the coordinate point?Remember that the quadrants are:
First quadrant: x > 0, y > 0.
Second quadrant: x < 0, y > 0.
Third quadrant: x < 0, y < 0.
Fourth quadrant: x > 0, y < 0.
In this case our coordinate pair is (0, -1/2).
So it will be between the third and fourth quadrants.
And yes, because one of the variables is zero, it is on the y-axis (just between the two quadrants).
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When you mix two colors of paint in equivalent ratios, the resulting color is always the same. Complete the table as you answer the questions.
How many cups of yellow paint should you mix with 1 cup of blue paint to make the same shade of green? Explain or show your reasoning.
Make up a new pair of numbers that would make the same shade of green. Explain how you know they would make the same shade of green.
row 1
cups of blue paint
cups of yellow paint
row 2
2
10
row 3
1
5
row 4
3
15
What is the proportional relationship represented by this table?
What is the constant of proportionality? What does it represent?
By Calculating the Constant of Proportionality, we get,
1. 5 cups of yellow paint is required for 1 cup of blue paint.
2. For 4 cups of blue paint, 20 cups of yellow paint is required.
For 5 cups of blue paint, 25 cups of yellow paint are required.
3. Cups of Yellow Paints = 5* Cups of Blue Paints
4. The constant of Proportionality is 5.
Let the cups of blue paint = x
and cups of yellow paint = y
From the table, we can infer that to make the same shade of green, we need to mix 5 cups of yellow color with 1 cup of blue.
We have, To make the same shade of green,
we need to mix 5 cups of yellow color with 1 cup of blue.
we need to mix 10 cups of yellow color with 2 cups of blue.
we need to mix 15 cups of yellow color with 3 cups of blue.
So, here we can see a relationship between the two colors, blue (x) and green(y)
Let k be the constant of proportionality, then, we have :
10 = k *2
k =[tex]\frac{10}{2} = 5[/tex]
Hence, For one cup of blue paint, we need 5 cups of yellow paint to make the same shade of green.
And the equation of the same is y =kx, that is y =5x....equation(1)
To make new pair of numbers that would make the same shade of green.
We can use the equation 1,
for x = 4, we need y = 20
foe x =5, we need y = 25
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Help i need an answer.
4. Manu determines the roots of a polynomial equation by applying the theorems he knows. He organizes the results of these theorems.
From the fundamental theorem of algebra, Manu knows there are 3 roots to the equation.
From Descartes’ rule of sign, Manu finds no sign changes in and 3 sign changes in .
The rational root theorem yields as the list of possible rational roots.
The lower bound of the polynomial is .
The upper bound of the polynomial is 1.
What values in Manu’s list of rational roots should he try in synthetic division in light of these findings?
The values in Manu’s list of rational roots he should try in the synthetic division, in light of these findings is
"Manu should try first +1/4,+1/2 and +1."
This is further explained below.
What is the synthetic division?Generally, Because the lower and higher bonds are located between -6 and 1, Manu should have only analyzed the potential rational zeros that fall between those two numbers.
A lower bond indicates that the feasible rational zero cannot be lower than -6, and therefore the higher number that may be achieved cannot be more than 1.
In this manner, Manu will get an opportunity to test. This approach of rationalization based on bonds aims to reduce the number of potential solutions.
Therefore, if Manu discovers through the synthetic division that the possible roots are +1/4,+1/2,+1,+5/4,+2,+5/2,+4,+5,+10,+20, then he should only consider those inside the intervals marked by the lower and upper bounds, which are +1/4,+1/2,+1, because the rest is greater than 1. This is because the rest of the possible roots are higher than 1.
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CQ
Manu determines the roots of a polynomial equation p(x)=0 by applying the theorems he knows. He organizes the results of these theorems. From the fundamental theorems of algebra, Manu knows there are 3 roots to the equation. From Descartes' rule of sign, Manu finds no sign changes in p(x) and 3 sign changes in p(-x). The rational root theorem yields +1/4,+1/2,+1,+5/4,+2,+5/2,+4,+5,+10,+20 as a list of possible rational roots. The lower bound of the polynomial is -6. The upper bound of the polynomial is 1. What values in Manu's list of rational roots should he try in synthetic division in light of these findings?
A kite is shown below. What is the length of LINE AC? (image attached)A. 4B. 6C. 9D. 10
The length of AC is given by AE + EC
According to the Pythagorean Theorem, we know that:
AE² = AD² - ED²
AE² = 5² - 3²
AE² = 25 - 9
AE² = 16
AE = 4
EC² = DC² + ED²
Since DC = BC, we have:
EC² = BC² + ED²
EC² = 45 - 9
EC² = 36
EC = 6
Therefore, AC = 4 + 6 = 10
Answer: D. 10
What is the length of a side of an equilateral triangle whose altitude is 5?
To make things easier to understand, let's see this figure, which represents the given situation:
According to this diagram, to find x, which is the length of the side of the triangle, we can use pythagorean theorem, this way:
[tex]\begin{gathered} x^2=(\frac{x}{2})^2+5^2 \\ x^2=\frac{x^2}{4}+25 \\ x^2-\frac{x^2}{4}=25 \\ \frac{3}{4}x^2=25 \\ x^2=\frac{25}{3}\cdot4^{} \\ x=\sqrt[]{33.3333} \\ x=5.77 \end{gathered}[/tex]plesehelppppppppppppppppppp!
Answer:
54 x 5
Step-by-step explanation:
the first figure out the brackets, then multiply by 5.
plss mark me as brainlist
Answer:
the answer is 6 ^ 15 you are very welcom
a box contains 3 white balls and 4 black balls. a ball is drawn at random the color is recorded and then the ball is put back in the box. Then a second ball is drawn at random from the same box. find the probability of the event that at least one of the balls is white
The box has 3 white balls and 4 black balls.
Total number of balls = 3 + 4 = 7
First draw:
The probability of getting a white ball is given by
[tex]\begin{gathered} P(white)=\frac{\text{number of white balls}}{total\text{ number of balls}} \\ P(white)=\frac{3}{7} \end{gathered}[/tex]Second draw:
Notice that after the first draw the ball is put back in the box.
The probability of getting a white ball is given by
[tex]P(white)=\frac{3}{7}[/tex]At least one of the balls is white means that one white ball or two white balls.
[tex]P(x\ge1)\; =P(x=1)+P(x=2)_{}[/tex]We have already found the probability of getting one white ball that is P(x=1) = 3/7
The probability of getting two white balls is
[tex]\begin{gathered} P(two\; white)=P(white)\times P(white) \\ P(two\; white)=\frac{3}{7}\times\frac{3}{7}=\frac{9}{49} \end{gathered}[/tex]Finally, the probability of at least one white ball is
[tex]\begin{gathered} P(x\ge1)\; =P(x=1)+P(x=2)_{} \\ P(x\ge1)\; =\frac{3}{7}+\frac{9}{49} \\ P(x\ge1)\; =\frac{30}{49} \end{gathered}[/tex]Therefore, the probability of the event that at least one of the balls is white is 30/49
The graph shows the reciprocal parent function.Which statement best describes the function?O A. The function is negative when x < 0.OB. The function is never negative.C. The function is negative when x > 0.O D. The function is negative when x < 0 and also when x > 0.PREVIOUS
Let's begin by identifying key information given to us in the graph:
The reciprocal parent function is given as 1/f(x). This is better written as shown below
[tex]\begin{gathered} f(x)=\frac{a}{(x-h)}+k \\ when\colon h=0,k=0,a=1 \\ f(x)=\frac{1}{x-0}+0 \\ f(x)=\frac{1}{x} \\ f(x)=y \\ \Rightarrow y=\frac{1}{x} \end{gathered}[/tex]When the value for x is greater than zero, the function is positive as shown below:
[tex]\begin{gathered} x=2 \\ y=\frac{1}{2}=\frac{1}{2} \\ y=\frac{1}{2} \end{gathered}[/tex]When the value of x is lesser than zero, the function is negative as shown below:
[tex]\begin{gathered} x=-1 \\ y=\frac{1}{-1}=-1 \\ y=-1 \end{gathered}[/tex]Therefore, the correct answer is option A (The function is negative when x < 0)
Question 2-22
A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eatin
5
36
35
36
O
●
cakes/week
1
01 cakes/week
35
1
1 cakes/week
4
< Previous
cakes/week
2022-2023 T-Math-Gr7Reg-T2-CBT: Section 2-...
Jake's weekly rate of eating of cake slices is 11.6
Given,
Number of equal slices of cake = 12
Number of slices Jake eaten after 3 days = 5
We have to find the weekly rate of eating;
Here,
Number of days in a week = 7
Jake eaten 5 slices in 3 days so, 7 - 3 = 4
Again after 3 days 5 slices.
Then,
4 - 3 = 1
That is, Jake eaten 10 slices of cake in 6 days.
Number of cake slices eaten in 1 day = 5/3 = 1.6
Therefore,
Weekly rate of eating is 10 + 1.6 = 11.6
That is,
Jake's weekly rate of eating of cake slices is 11.6
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A triangle, ABC, has angle measures of 45°, 45°, and 90° and two congruent (equal) sides. How would this triangle be classified?
O Isosceles acute
O Scalene acute
O Scalene right
O Isosceles right
Answer:
(d) Isosceles right
Step-by-step explanation:
You want to know the classification of a triangle with angles 45°-45°-90°.
ClassificationThe largest angle is 90°, a right angle. Any triangle containing a right angle is a right triangle.
Two of the angles have the same measure. This means the triangle is an isosceles triangle.
The triangle is classified as an isosceles right triangle.
__
Additional comment
A triangle whose largest angle is less than 90° is an acute triangle. If the largest angle is greater than 90°, it is an obtuse triangle.
If no sides are the same length, the triangle is scalene. If two sides are the same length, the triangle is isosceles. If all three sides are the same length, it is an equilateral triangle.
Larger angles are opposite longer sides, so if angles are the same, their opposite sides are the same.
It can be worthwhile to remember that the side length ratios in an isosceles right triangle are 1 : 1 : √2.
i need help with this asap please
Answer:
Given that,
[tex]2+(-8)+32+(-128)+.\ldots_{}[/tex]To find the sum of the first 5 terms.
First, to find the first 5 terms of the given sequence.
The given sequence is 2,-8,32,-128,...
It follows geometric series with initial term 2, and common ratio as -4
The explicit formula of the given sequence is,
[tex]t_n=2(-4)^{n-1}_{}_{}[/tex]To find the 5th term of the sequence,
Put n=5 in the above equation we get,
[tex]t_5=2(-4)^{5-1}[/tex][tex]t_5=2(-4)^4[/tex][tex]t_5=2(256)[/tex][tex]t_5=512[/tex]Since common ratio is less than 1, we get the sum of the series formula as,
[tex]S_n=\frac{a(1-r^n)}{1-r}[/tex]Substituting the values we get,
[tex]S_5=\frac{2(1-(-4)^5)}{1+4}[/tex][tex]=\frac{2(1+1024)}{5}[/tex][tex]=\frac{2(1025)}{5}[/tex][tex]=2(205)[/tex][tex]=410[/tex]The sum of the first 5 terms of the given series is 410.
Answer is: option B: 410
A car loan totaling 13,456.44 paid off in 36 equal monthly payment can afford no more than $350 per month can she or down explain
She cannot afford to buy the car with monthly instalments of $350. She needs to make a downpayment of $856.44 to afford the car
The total amount of the car loan = 13456.44
Number of instalments = 36 monthly payments
Affordable installment per month = $350
EMI: A fixed payment is made by a borrower to a lender at a specific time each month, known as an equated monthly instalment (EMI). The loan is repaid in full over a predetermined period of time by making equal monthly payments to the principal and interest.
Total payment that can be made using the instalments = Monthly instalments * Number of instalments
= 350*36
Total payment that can be made using the instalments = 12600
The amount remaining after paying instalments = 13456.44 - 12600
= 856.44
Downpayment required = $856.44
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Find the area of a triangular window with the given base and height base = 10 ft height =7 ft
The area of a triangular shape is given by the following formula
[tex]A=\frac{bh}{2}[/tex]where,
base, b = 10 ft
height, h = 7 ft
therefore,
[tex]A=\frac{10\cdot7}{2}=\frac{70}{2}=35[/tex]thus, the answer is 35 ft^2
What is the slope of the line passing through (-2, 4)
and (3, -4)?
Answer:
[tex]\frac{-8}{5}[/tex]
Step-by-step explanation:
Slope is the change in y over the change in x
[tex]\frac{y_{2 - y_{1} } }{x_{2 - y_{1} } }[/tex]
(-2,4) is ([tex]x_{1}[/tex], [tex]y_{1}[/tex])
(3. -4) is ([tex]x_{2}[/tex],[tex]y_{2}[/tex])
[tex]\frac{-4 - 4}{3 - -2}[/tex] = [tex]\frac{-8}{3+ 2}[/tex] = [tex]\frac{-8}{5}[/tex]
Answer:
-8 / 5
Step-by-step explanation:
We can use the slope formula
m = ( y2-y1)/(x2-x1)
= ( -4 -4)/( 3 - -2)
= -8/ ( 3+2)
= -8 / 5
(3)/(4)[(-15+4)+(6+7)-:(-3)]
Answer:
The answer is -23/2 or -11.5
Step-by-step explanation:
Question 1. Find the center and radius of the circunscribed circle.
In order to find the center of the circunscribed circle, we can use the midpoint theorem because the center point is in the middle of any two vertices
that is, if we take points (9,23) and (8,16) the midpoint C is given as
[tex]C=(\frac{8+9}{2},\frac{16+23}{2})[/tex]which gives
[tex]C=(8.5,19.5)[/tex]So the center of the circle is the point (8.5,19.5)
On the other hand, the radius is equal to the distance from any vertex to the center. If we take the vertex (8,16), we get
[tex]r=\sqrt[]{(8.5-8)^2+(19.5-16)^2}[/tex]which gives
[tex]\begin{gathered} r=\sqrt[]{0.5^2+3.5^2} \\ r=\sqrt[]{0.25+12.25} \\ r=\sqrt[]{12.5} \\ r=3.5355 \end{gathered}[/tex]so, the radius measure 3.54 units.
Now, lets prove that the answer are correct. In order to do that, we can choose the other vertices and apply the same procedure as above.
So the vertices are (5,20) and (12,19). Again, the center is the midpoint between these points and is given as
[tex]\begin{gathered} C=(\frac{5+12}{2},\frac{20+19}{2}) \\ C=(\frac{17}{2},\frac{39}{2}) \\ C=(8.5,19.5) \end{gathered}[/tex]which is the same center as above.
Now, the distance from the center to vertec (5,20) is
[tex]\begin{gathered} r=\sqrt[]{(8.5-5)^2+(20-19.5)^2} \\ r=\sqrt[]{3.5^2+0.5^2} \\ r=\sqrt[]{12.25+0.25} \\ r=\sqrt[]{12.5} \\ r=3.5355 \end{gathered}[/tex]which is the same radius obtained above. Then, the answers are correct.
Which of the following is the correct mathematical expression for:
The difference between three times a number and 4
A. 3x + 4
B. 3x - 4
C. 1/3x + 4
D. 1/3x - 4
Answer:
B
Step-by-step explanation:
if the number is x then 3 times the number is 3x
the difference is the subtraction of 3x and 4 , that is
3x - 4
Study the spinner below. If the wedges 2 and 5 are twice the area of the otherwedges what is the likelihood of landing on 4 or 6?23416/5A.NIO.B.A-OC.100oD. There is not enough information provided.
As we can see, there are 8 parts in our circle but 4 parts belong to wedge 2 and 5. Then, the probability of landing on 4 or 6 is given by
[tex]P(4\cup5)=P(4)+P(5)[/tex]because part 4 and part 5 are disjoint sets. Since there are 8 parts in our circle, we have
[tex]\begin{gathered} P(4\cup5)=P(4)+p(5) \\ P(4\cup5)=\frac{1}{8}+\frac{1}{8} \end{gathered}[/tex]which gives
[tex]P(4\cup5)=\frac{2}{8}=\frac{1}{4}[/tex]which corresponds to option B
Pls help if someone can solve for me 7-15. Thank you!
Given:
Set U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
Set P: {2, 3, 5 7, 9, 11, 13, 17, 19}
Questin 7.
Complement of set P.
The complement of set P will be the set that contains the elements present in set U but not in P.
Hence, we have:
ANSWER:
P' = {1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20}
Select all the intervals where if is increasing
The given function is increasing over these following intervals:
B. -2 ≤ x ≤ -1.
C. 0 < x < 1.
Where to find where a function is increasing, from it's graph?Considering the graph of the function, we get that a function f(x) is increasing when it is "moving northeast", that is, to the right and up on the graph, meaning that when x increases, y increases.Conversely, we get that a function f(x) is decreasing when it is "moving southeast", that is, to the right and down the graph, meaning that when x increases, y decreases.Considering the given definitions on the bullet points above, the behavior of the function can be separated as follows:
Increasing on -2.5 ≤ x ≤ 2.5.Decreasing on all other intervals except the one above.Items B and C have subsets of the interval -2.5 ≤ x ≤ 2.5, hence the function is increasing on those intervals, and they are the correct options for this problem.
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The nutty professor sells cashews for $6.20 per pound and Brazil nuts for $4.30 per pound. How much of each type should be used to make a 32 pound mixture that sells for $5.37 per pound?
Let's call C the weight of the Cashews and B the weight of the Brazil nuts.
Since the professor wants to sel a 32 pound mixture, we have the following equation:
[tex]C+B=32.[/tex]Now, since the price of the mixture will be $5.37 per pound, this means the price of the whole mixture will be
[tex]32\cdot5.37=171.84.[/tex]This leads us to the followin equation:
[tex]6.20C+4.30B=171.84.[/tex]Now we have a system of equations:
[tex]C+B=32[/tex][tex]6.20C+4.30B=171.84.[/tex]To solve it, let's isolate one of the variables in the first equation by subtracting B from both sides of it:
[tex]C=32-B\text{.}[/tex]Now, let's use this value of C in the second equation:
[tex]6.20(32-B)+4.30B=171.84,[/tex][tex]198.4-6.20B+4.30B=171.84,[/tex][tex]198.4-1.9B=171.84.[/tex]To solve this equation, let's subtract 198.4 from both sides of it:
[tex]-1.9B=-26.56.[/tex]Then, let's divide both sides by -1.9:
[tex]B\approx13.98.[/tex]Using this value of B in the first equation will give us:
[tex]C=32-13.98=18.02.[/tex]These would be the values of B and C round to two decimals. If we want, we can also write them as integer numbers. so the mix would need to have 18 pounds of cashews and 14 pounds of Brazil nuts.
There are 2.54 centimeters in 1 Inch. There are 100 centimeters in 1 meter. To the nearest inch, how many inches are in 7 meters? Enter the answer in the box. inches
Based on the given equivalences, you have:
7 m = 7(100 cm) = 700 cm
700 cm = 700 (2.54 in) = 1,778 in
Hence, there are 1,778 inches in 7 meters
Which of the following sets of numbers could represent the three sides of a righttriangle?Submit AnswerO {9, 12, 14}O {48, 55,73}O {11, 59, 61}{8, 40, 41}
Answer:
{48, 55,73}
Explanation:
To determine if any three side lengths can form a right triangle, we check if it satisfies the Pythagorean Theorem.
[tex]\begin{gathered} c^2=a^2+b^2 \\ c\text{ is the longest side, the hypotenuse} \end{gathered}[/tex][tex]\begin{gathered} 14^2=9^2+12^2 \\ 196\neq225 \end{gathered}[/tex][tex]\begin{gathered} 73^2=48^2+55^3 \\ 5329=5329 \end{gathered}[/tex]Therefore, the side lengths that could form a right triangle are {48, 55,73}.
Find sin(2x), cos(2x), and tan(2x) from the given information.
tan(x) = − 4/3 , x in Quadrant II
Answer:
Step-by-step explanation:
Starting from tan x, you can find sec x, because of the trigonometric identity 1 + tan2 x = sec2 x.
1 + (-4/3)2 = 1 + 16/9 = 25/9 = sec2 x.
So sec x = ±√(25/9) = ±5/3. But since x is in Quadrant II, sec x has to be negative. That's because sec x has the same sign as cos x, because sec x = 1 / cos x. We know that cos x is negative is Quadrant II, therefore so is sec x. So sec x = -5/3.
Since sec x and cos x are reciprocals of each other, cos x = 1/sec x = -3/5.
Now use the identity sin2 x + cos2 x = 1, to find sin x:
sin2 x + (-3/5)2 = 1
sin2 x + 9/25 = 1
sin2 x = 16/25
sin x = ±4/5
Again, we know that sin x is positive in Quadrant II, so sin x = 4/5.
Now that we know sin x and cos x, we can use the double angle formulas to find sin 2x and cos 2x.
sin 2x = 2 sin x cos x = 2 (4/5) (-3/5) = -24/25
cos 2x = cos2 x - sin 2 x = (-3/5)2 - (4/5)2 = 9/25 - 16/25 = -7/25
Finally use the identity tan x = sin x / cos x to find tan 2x:
tan 2x = sin 2x / cos 2x = (-24/25) / (-7/25) = -24/-7 = 24/7
The National Opinion Research Center administered the General Social Surve
persons in the United States who were 18 years of age or older. One question asked
respondents for their highest grade of school completed (Respondent's Years of Education).
Another question asked respondents for the highest grade of school that their father had
completed (Father's Years of Education).
The equation for the least squares regression line for predicting a respondent's years of
education, ý, from the years of education for the respondent's father, x, is:
ý = 9.996 + 0.355x
Predict the years of education for a person whose father had 14 years of education. Round to
the nearest whole number.
The years of education as per the given equation, for a person whose father had 14 years of education is 14.966 years.
What is education?
A planned activity, education has objectives like knowledge transmission or character and skill development. The development of understanding, reason, kindness, and honesty may be some of these goals.
As given in the question,
respondent's years of education is represented by y, and
respondent's father years of education is represented by x,
The equation for the least squares regression line for predicting a respondent's years of education is given as:
y = 9.996 + 0.355x
The year of education of respondent's father is given as 14
So putting the value of x in the given equation as 14:
we get,
y = 9.996 + 0.355(14)
y = 14.966
Hence, the years of education for a person whose father had 14 years of education is 14.966 years.
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During a basketball game, you made 11 shots of either 2 or 3 points. You scored atotal of 25 points. How many shots of each point value did you make?
During a basketball game, you made 11 shots of either 2 or 3 points. You scored a
total of 25 points. How many shots of each point value did you make?
Let
x -----> number of shots of 2 points
y -----> number of shots of 3 points
we have that
x+y=11
x=11-y -----> equation A
and
2x+3y=25 -----> equation B
substitute equation A in equation B
2(11-y)+3y=25
solve for y
22-2y+3y=25
y=25-22
y=3
Find the value of x
x=11-3
x=8
therefore
number of shots of 2 points is 8number of shots of 3 points is 3