Using the Factor Theorem, the polynomials are given as follows:
1. [tex]P(x) = x^5 + 2x^4 - 7x^3 + x^2[/tex]
2. [tex]P(x) = 0.8(x^4 - 4x^3 - 16x^2 + 64x)[/tex]
3. P(x) = -0.1(x³ - 4x² - 3x + 18)
What is the Factor Theorem?The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
In which a is the leading coefficient.
Item a:
The parameters are:
[tex]a = 1, x_1 = x_2 = 1, x_3 = x_4 = 0, x_5 = -4[/tex]
Hence the equation is:
P(x) = (x - 1)²x²(x + 4)
P(x) = (x² - 2x + 1)(x + 4)x²
P(x) = (x³ + 2x² - 7x + 1)x²
[tex]P(x) = x^5 + 2x^4 - 7x^3 + x^2[/tex]
Item b:
The roots are:
[tex]x_1 = x_2 = 4, x_3 = 0, x_4 = -4[/tex]
Hence:
P(x) = a(x - 4)²x(x + 4)
P(x) = a(x² - 16)x(x - 4)
P(x) = a(x³ - 16x)(x - 4)
[tex]P(x) = a(x^4 - 4x^3 - 16x^2 + 64x)[/tex]
It passes through the point x = 5, P(x) = 36, hence:
45a = 36.
a = 4/5
a = 0.8
Hence:
[tex]P(x) = 0.8(x^4 - 4x^3 - 16x^2 + 64x)[/tex]
Item 3:
The roots are:
[tex]x_1 = x_2 = 3, x_3 = -2[/tex]
Hence:
P(x) = a(x - 3)²(x + 2)
P(x) = a(x² - 6x + 9)(x + 2)
P(x) = a(x³ - 4x² - 3x + 18)
For the y-intercept, x = 0, y = -1.8, hence:
18a = -1.8 -> a = -0.1
Thus the function is:
P(x) = -0.1(x³ - 4x² - 3x + 18)
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Simplify x^2+2x+1 divided by x+1
The quotient is x+1.
What Long division method?Long division is a method for dividing one large multi digit number into another large multi digit number.
Given:
f(x)= x²+2x+1
g(x)= x+1
The steps to divide the polynomial is :
Divide the leading term of the dividend by the leading term of the divisor: x²/x=x.Multiply it by the divisor: x(x+1)=x²+x.Subtract the dividend from the obtained result: (x²+2x+1)−(x²+x)=x+1.Divide the leading term of the obtained remainder by the leading term of the divisor: x/x=1.Multiply it by the divisor: 1(x+1)=x+1.Subtract the remainder from the obtained result: (x+1)−(x+1)Learn more about long division method here:
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From this diagram, select the
pair of lines that must be
parallel if angle 2 is congruent to angle 6. If there is no pair of lines, select none
Answer:
A). O is parallel to qif angle 2 is congruent to 6 them the slope of line o and q is equal
that is line o is parallel to q
Translate each English phrase in the following problem into an algebraic expression and set up the related equation. Let z be the unknown number. The difference between a number and −23 − 23 is equal to the product of the number and 13 13 . Step 2 of 3 : Translate "the product of the number and 13 13 ".
The English phrase of the obtained equation is 12 times z subtracted by 3. The obtained equation after solving is 12z -23
What exactly is simplification?Simplifying means making something easier to do or comprehend, as well as making something less difficult.
Given data;
z be the unknown number
Given conditions;
1. The difference between a number and −23.
2.−23 is equal to the product of the number and 13.
The mathematical form of the given phrase is;
⇒ z - (-23) = z × 13
⇒z+23 = 13z
⇒12z -23
The English phrase of the obtained equation is 12 times z subtracted by 3.
Hence the simplification of the given expressions is 12z -23.
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What is the point of intersection of the lines represented by these equations
Jy = -2x + 2
y = x - 4
A. (-2, 2)
B. (1, -3)
c. (2,-2)
D. (3,-4)
To find the point of intersection of two lines, we can use substitution to find the x and y coordinates.
Solving the QuestionWe're given:
[tex]y = -2x + 2[/tex][tex]y = x - 4[/tex]Use substitution to find x:
[tex]-2x + 2=x-4\\-2x=x-6\\-3x=-6\\x=2[/tex]
Use substitution to find y:
[tex]y=x-4\\y=2-4\\y=-2[/tex]
AnswerThe point of intersection is (2,-2).
Given that the slope is 4 and (4, 6) is on the line. Write the equation of the line
in the slope-intercept
form
Answer:
y= 4x - 10
Step-by-step explanation:
y-6 = 4(x - 4)
y - 6 = 4x -16
+6. +6
y= 4x - 10
Describe the end behavior for the graph of y=-x³- ax + 3, if a is a constant.
Answer: Choice C
As [tex]\text{x} \to \infty[/tex] then [tex]\text{y} \to -\infty[/tex]
As [tex]\text{x} \to -\infty[/tex] then [tex]\text{y} \to \infty[/tex]
Check out the diagram below.
========================================================
Explanation:
The leading term is the term with the largest exponent. That would be the term of [tex]-\text{x}^3[/tex]. The leading term directly determines the end behavior. The other terms will not play a role.
What this means is that the end behavior of [tex]\text{y} = -\text{x}^3 - \text{ax} + 3[/tex] is the exact same as the end behavior of [tex]\text{y} = -\text{x}^3[/tex]
As x gets really big in the positive direction, we'll have [tex]-\text{x}^3[/tex] get really small in the negative direction. For instance, x = 10 leads to [tex]-\text{x}^3 = -10^3 = -1000[/tex] and x = 100 leads to [tex]-\text{x}^3 = -100^3 = -1,000,000[/tex]
Therefore, as [tex]\text{x} \to \infty[/tex] then [tex]\text{y} \to -\infty[/tex]
Visually the graph goes forever downward when we move to the right. We can consider this as "falls to the right".
--------------------------------
The graph "rises to the left" because as [tex]\text{x} \to -\infty[/tex] then [tex]\text{y} \to \infty[/tex]
Here's a small table to give numeric examples:
[tex]\begin{array}{|c|c|} \cline{1-2}x & y\\\cline{1-2}-1 & 1\\\cline{1-2}-10 & 1000\\\cline{1-2}-100 & 1,000,000\\\cline{1-2}\end{array}[/tex]
As x gets more negative, y becomes more positive. We have this opposite nature going on similar to the previous section.
This graph goes uphill when moving to the left.
Check out the graph below. In the diagram I used a = 2, but you could use any value of 'a' you want to get the same end behavior.
The power 3 Superscript negative 3 equals StartFraction 1 Over 27 EndFraction . Which expression is equivalent to 3 Superscript negative 3?
Applying the negative exponent, it is found that the equivalent expression to [tex]3^{-3}[/tex] is [tex]\frac{1}{27}[/tex].
How to deal with a negative exponent?When we have a negative exponent, we use a fraction, with the term with the exponent going to the denominator, as follows:
[tex]a^{-b} = \frac{1}{a^b}[/tex]
Hence, in this problem, the equivalent expression is:
[tex]3^{-3} = \frac{1}{3^3} = \frac{1}{27}[/tex]
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Which expression is modeled by this arrangement of tiles?
40 positive tiles are broken up into 10 groups of 4 positive tiles.
The answer is 40/10 because the total number of square units is 40 option second 40/10 is correct.
What is a fraction?Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.
The question is incomplete.
The complete question is:
Which expression is modeled by this arrangement of tiles? 40 positive tiles are broken up into 10 groups of 4 positive tiles.
40/(-10) 40/10 40/40 40/(-40)Please refer to the attached picture.
We have a rectangular shape as shown in the picture.
Total number of square units = 40
Each row has 10 units cube.
Each cube = 40/10
Thus, the answer is 40/10 because the total number of square units is 40 option second 40/10 is correct.
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Find the slope and distance.
(6,-6) and (-6,+8)
Answer:
[tex]\rm slope: -\dfrac{7}{6}[/tex]
[tex]\rm Distance: 2\sqrt{85} \ units[/tex]
Part A[tex]\sf slope: \dfrac{y_2 - y_1}{x_2- x_1} = \dfrac{\triangle y }{\triangle x} \ \ where \ (x_1 , \ y_1), ( x_2 , \ y_2) \ are \ points[/tex]
Here given points are: (6, -6), (-6, 8)
Insert the values:
[tex]\sf \rightarrow slope: \dfrac{8-(-6)}{-6-6}[/tex]
[tex]\sf \rightarrow slope: \dfrac{8+ 6}{-12}[/tex]
[tex]\sf \rightarrow slope: \dfrac{14}{-12}[/tex]
[tex]\sf \rightarrow slope: -\dfrac{7}{6}[/tex]
Part B[tex]\sf Distance \ between \ points : \sf d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}[/tex]
Using the formula,
[tex]\sf d = \sqrt{(-6 - 6)^2 + (8-(-6))^2}[/tex]
[tex]\sf d = \sqrt{(-12)^2 + (14)^2}[/tex]
[tex]\sf d = \sqrt{144 +196}[/tex]
[tex]\sf d = \sqrt{340}[/tex]
[tex]\sf d = \sqrt{85 \times 4}[/tex]
[tex]\sf d = 2\sqrt{85}[/tex]
Hence, the distance is 2√85 units.
Hii!
___________________________________________________________
[tex]\stackrel\bigstar{\rightsquigarrow\circ\boldsymbol{\underbrace{Answer}\circ\leftharpoonup}}[/tex]
Slope =
Distance =
[tex]\stackrel\bigstar{\rightsquigarrow\circ\boldsymbol{\underbrace{Explanation}\circ\leftharpoonup}}[/tex]
[tex]\bullet[/tex] Let's work out the slope
[tex]\twoheadrightarrow\sf \cfrac{y2-y1}{x2-x1}[/tex]
[tex]\bullet[/tex] Stickin the values
[tex]\twoheadrightarrow\sf \cfrac{8-(-6)}{-6-6}[/tex]
[tex]\bullet[/tex] Simplify
[tex]\twoheadrightarrow\sf \cfrac{8+6}{-12}[/tex]
[tex]\bullet[/tex] Finish simplifying
[tex]\twoheadrightarrow\sf \cfrac{14}{-12}[/tex]
[tex]\bullet[/tex] Reduce the fraction
[tex]\twoheadrightarrow\sf \cfrac{7}{-6}[/tex]
____________________________________[tex]\bullet[/tex] Work out the distance
[tex]\twoheadrightarrow\sf \sqrt{(x2-x1)^2+(y2-y1)^2}[/tex]
[tex]\bullet[/tex] Stick in the values
[tex]\twoheadrightarrow\sf \sqrt{(-6-6)^2+(8-(-6)^2}[/tex]
[tex]\bullet[/tex] Simplify the radical expression
[tex]\twoheadrightarrow\sf \sqrt{(-12)^2+14^2}\\\\\twoheadrightarrow\sf \sqrt{144+169} \\\\\twoheadrightarrow\sf \sqrt{340}\\\\\twoheadrightarrow\sf \sqrt{85\cdot4} \quad (simplifying\,the\,surd)[/tex]
[tex]\ddagger[/tex] [tex]\boldsymbol{DISTANCE=2\sqrt{85} \,Units}[/tex] [tex]\ddagger[/tex]
--
Hope that this helped! Best wishes.
--
Solve the inequality
Of the following fractions3/7, 5/8, 7/13, and 11/19 Which is the largest
Answer:
5/8
Step-by-step explanation:
5/8=0.6.25
0.625> all the other numbers
I need this answered fast pleasee
Suppose you and your friends form a band and you want to record a demo. Studio A rents for $75 plus $75 an hour and Studio B rents for $150 and $50 an hour. Let t = the number of hours and c = cost.
Part A Write a system of equations to represent the cost at each studio.
Part B Solve the system of equations from Part A. Explain your method.
Part C Explain what the solution to the system means in terms of where your band will rent a studio.
The descriptive answer of different part is described below.
What is Linear Equation?An equation that has the highest degree of 1 is known as a linear equation. This means that no variable in a linear equation has an exponent more than 1. The graph of a linear equation always forms a straight line.
Here, Studio A rents for $75 plus $75 an hour .
Studio B rents for $150 and $50 an hour.
Let t = the number of hours and c = cost.
Part A
Studio A: c = 75 +75·t
Studio B: c = 150 +50·t
Part B
We solve the system of equation using substitution method, where we substitute c in the first equation with 150 +50t.
150 +50·t = 75 +75·t, subtract 75 and 50t from both sides
150 -75 = 75t -50t, combine likes terms
75 = 25t, divide both sides by 25
3 = t
Now we know that t = 3 so we find c by substituting t with 3 in the second equation.
c = 150 +50·t
c = 150 +50·3
c = 300
The solution of the system of equations is c = 300 and t = 3
Part C:
The solution of the system of equation tells us that if we rent either studio A or B for 3 hours will pay the same price $300.
The price will be different if we rent studio A then studio B if we rent it for a different amount of time than 3 hours.
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5. Dan is making the perfect shade of purple paint for his Nana. The perfect shade of purple mixes red and blue paint at a
ratio of 4 to 5. Accidently Dan mixed five cups of red and four cups of blue, using up all his red paint. How much blue must
he add to fix his mistake?
Answer:
2 and 1/4 cups
Step-by-step explanation:
4 cups-5 cups
5 cups-?
(5×5)÷4=6 cups and 1/4
6 and 1/4-4=2 cups and 1/4
Suppose that in the country of Workanda, the population is 330 million, and 170 million people are working (have a job). If 70 million people are ineligible to work and the Labor Force Participation Rate is 70%, what is the unemployment rate in Workanda? [Use two significant digits for your answer]. A BIU - PE III Suppose that in the country of Workanda , the population is 330 million , and 170 million people are working ( have a job ) . If 70 million people are ineligible to work and the Labor Force Participation Rate is 70 % , what is the unemployment rate in Workanda ? [ Use two significant digits for your answer ] .
Answer:
30%
Step-by-step explanation:
The unemployment rate is 30%
look at the picture
A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.77 hours. Assume that a random sample of 40 mechanics is selected and the mean rebuild time of the sample is computed. Assuming the rebuilding times are normally distributed, what percentage of sample means are greater than 8.8 hours? Express your answer as a percent rounded to hundredths of a percent.
Answer:
40.9 %
Step-by-step explanation:
8.8 is .4 hours above the mean this is .4/1.77 = .226 standard deviations above the mean
Using z-score table this corresponds to ~ .5910 (59.10%)
this percentage is LESS than 8.8 hours
the percentage ABOVE this is 40.9%
(Not asked, but this is 40.9 (40 ) = ~ 16 of the 40 mechanics sampled)
Please help!!
Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the equation
(-4,5) y=1/3x +6
Answer:
The Equation of the line is y = [tex]\frac{-1}{3}[/tex]x + [tex]\frac{11}{3}[/tex]
Step-by-step explanation:
Multiplication of slope of two perpendicular lines is = -1
first line is y= [tex]\frac{1}{3}[/tex]x +6
slope of this line is [tex]\frac{1}{3}[/tex]
So the slope of the second line is : [tex]\frac{-1}{3}[/tex]
Equation of line would be written by using :
(y - y1) = m(x - x1)
here the line passing through (-4,5)
put the values in equation
y -5 =[tex]\frac{-1}{3}[/tex][x - (-4)]
3y - 15 = -x -4
3y = -x +11
y = [tex]\frac{-1}{3}[/tex]x + [tex]\frac{11}{3}[/tex]
So The Equation of the line is y = [tex]\frac{-1}{3}[/tex]x + [tex]\frac{11}{3}[/tex]
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calculate the median for: 5, 10, 12, 4, 6, 11, 13, 5r
Step-by-step explanation:
please mark me as brainlest
Answer:
8
Step-by-step explanation:
We need to sort your set into Least-to-Greatest order.
4, 5, 5, 6, 10, 11, 12, 13
That is your set in least to greatest order.
Which 2 numbers are in the middle?
They are 6 and 10.
Now, we find the mean of 6 and 10.
(6+10)/2 = 8
8 is your median
Question 1 of 10
Solve the rational equation:
2x/ x - 3 = 3x/ x - 2
On Friday, a local hamburger shop sold a combined total of 396 hamburgers and cheeseburgers. The number of cheeseburgers sold was two times the number
of hamburgers sold. How many hamburgers were sold on Friday?
Answer:792
Step-by-step explanation:
A water wheel is designed in the shape of a regular octagon. What is the perimeter of the water wheel?
The graph shows an eight-sided polygon. Some major points are (0, 4), (3, 3), (4, 0), (3, minus 3), (0, minus 4), (minus 3, minus 3), (minus 4, 0), and (minus 3, 3).
A.
B.
C.
D.
Perimeter is the sum of the length of the sides used to make the given figure. The perimeter of the regular octagon is 8√58 units.
What is the perimeter?Perimeter is the sum of the length of the sides used to make the given figure.
For a regular octagon, the length of each side of the octagon is the same. Therefore, the length of a single side of the octagon is,
Length of the side of the octagon = √[(-4-3)² + (0 + 3)²] = √58
Now, the perimeter of the octagon is 8 times of√58.
Perimeter = 8 ×√58 = 8√58
Hence, the perimeter of the regular octagon is 8√58 units.
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Question 4tiple Choice Worth 5 points)
(07.03 MC)
Circle A has center of (6, 7) and a radius of 4, and circle B has a center of (2, 4) and a radius of 16. What steps will help show that circle A is similar to circle B?
O Translate circle A using the rule (x+4y+3)
O Rotate circle A 45" about the center
O Dilate circle A by a scale factor of 4
O Reflect circle A about the origin
Answer:
Dilate circle A by a scale factor of 4.
Step-by-step explanation:
To show that two circles are similar, use transformations to map one to the other.
Step 1
Translate circle A so that its center is the same as circle B.
Center of circle A = (6, 7)Center of circle B = (2, 4)Translate circle A using the rule (x - 4, y - 3)
Step 2
Find the scale factor needed to dilate circle A to the size of circle B using the radii of the circles.
Radius of circle A = 4Radius of circle B = 16⇒ scale factor = 16 ÷ 4 = 4
Dilate circle A by a scale factor of 4.
Therefore, the only answer option that is correct is:
Dilate circle A by a scale factor of 4.
NEED HELP ASAP WILL MARK BRAINLIEST!
Answer:
1.5
Explanation:
Given that:
[tex]\sf log_{b} (3) = 1.099[/tex][tex]\sf log_{b} (5) = 1.609[/tex]While solving this problem, the above following is not required to use.
[tex]\rightarrow \sf log_b ([b^3])^{1/2}[/tex]
apply log rule
[tex]\rightarrow \sf \dfrac{1}{2}\log _b\left(b^3\right)[/tex]
[tex]\boxed{\sf \log _b\left(b^3\right)=3}[/tex]
[tex]\rightarrow \sf \dfrac{1}{2}(3)[/tex]
distribute
[tex]\rightarrow \sf \dfrac{3}{2} \quad or \quad 1.5[/tex]
What is the slope-intercept equation for the line below?
10
(0, 2)
(5, 4)
Answer:slope is 0.4
Step-by-step explanation:
y2-y1 divided by x2-x1
2-4/0-5=0.4
Find the surface area of the composite figure
4 cm
13 cm
3 cm
4 cm
12 cm
SA = [?] cm²
5 cm
3 cm. Pls help
Two owners of a cattle ranch, Robert and Val, want to find the average weight for the ranch’s 200 cows. Instead of weighing all of the cows:
1. Robert weighs 25 cows and gets an average weight of 1,550 pounds (stdev = 50)
2. Val weighs 100 cows and gets an average weight of 1,380 pounds (stdev = 50)
The margin of error will be 20 and 10 respectively.
How to calculate the margin of error?The margin of error for Robert will be:
= 1.96 × 50/✓25
= 1.96 × (50/5)
= 1.96 × 10
= 20 approximately
The margin of error for Val will be:
= 1.96 × 50/✓100
= 1.96 × 50/10
= 1.96 × 5
= 10 approximately
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Which statement describes how the y-intercept of the graph of f(x) = |2x − 1| compares to the y-intercept of the graph of g(x) = |2x − 5|? A. The y-intercept of g is 4 units above the y-intercept of f. B. The y-intercept of g is 4 units below the y-intercept of f. C. The y-intercept of g is 4 units to the right of the y-intercept of f. D. The y-intercept of g is 4 units to the left of the y-intercept of f.
The statement "the y-intercept of g is 4 units above the y-intercept of f'' is true option first is correct.
What is a function?
It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
We have two functions:
f(x) = |2x - 1| and
g(x) = |2x - 5|
After plotting on a coordinate plane.
The y-intercept of f(x) is 1
The y-intercept of g(X) is 5
Or plug x = 0 to find the y-intercept.
The y-intercept of g is 4 units above the y-intercept of f
Thus, the statement "the y-intercept of g is 4 units above the y-intercept of f'' is true option first is correct.
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Caitlyn and Steve are planning their wedding reception and must decide between two catering halls. The first site, A Wedding Space, rents for $900 for one day and charges $50 per person for dinner. The second venue, A Wedding Place, costs $450 to rent for one day and charges $75 per person for the same dinner. Solve the equation 900+50x=450+75x to determine how many guests they can invite so that the cost they pay will be the same at both wedding catering halls.
The guest they can invite so that the cost they pay will be the same at both wedding catering halls is 18.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
The given equation is
900+50x=450+75x
To solve;
900 - 450 = -50x +75x
450 = 25x
x = 450 /25
x = 18
The guest they can invite so that the cost they pay will be the same at both wedding catering halls is 18.
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HELPPPPPPP
What are the solutions to the system of equations graphed below?
A. (2,0) and (0,-4)
B. (-2,0) and (2,0)
C. (0,-2) and (0,2)
D. (0,2) and (-4,0)
Answer:
It's A. (2, 0) and (0, -4)
Question 4: 11 pts
Elena wants to earn at least $180 a week to help her parents with bills. She determines that
she can charge $15 an hour to babysit and can earn $10 an hour bagging groceries. She can
only work 13 hours per week. Write and graph a system of linear inequalities that shows the
number of hours she can work so she can help her parents. Use x for the number of hours
babysitting and y for the number of hours bagging groceries.
our system of inequalities is:
x + y ≤ 13.
x*$15 + y*$10 ≥ $180.
The graph can be seen at the end.
How to get the system of inequalities?Here we have the variables:
x = number of hours babysitting.y = number of hours bagging groceries.We know that she can only work 13 hours per week, then we have:
x + y ≤ 13.
And she wants to earn at least $180 per week, then:
x*$15 + y*$10 ≥ $180.
So our system of inequalities is:
x + y ≤ 13.
x*$15 + y*$10 ≥ $180.
The graph of the system can be seen below. The solution set is defined by the zone where the two shaded regions intersect (where we also should add the restrictions x >0, y > 0).
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