Answers
Answer:
f(x) = x³ -4x² +9x +164
Step-by-step explanation:
When a function has a zero at x=p, it has a factor (x-p). When a polynomial function with real coefficients has a complex zero, its conjugate is also a zero.
Factored formGiven the two zeros and the one we can infer, we can factor our 3rd-degree polynomial function as ...
f(x) = a(x -(-4))·(x -(4+5i))·(x -(4-5i))
Real factorsUsing the factoring of the difference of squares, we can combine the complex factors to make a real factor.
f(x) = a(x +4)((x -4)² -(5i)²) = a(x +4)(x² -8x +16 +25)
Finding the scale factorThe value of this at x=1 is ...
f(1) = a(1 +4)(1 -8 +41) = 170a
We want f(1) = 170, so ...
170 = 170a ⇒ a=1
The factored polynomial function is ...
f(x) = (x +4)(x² -8x +41)
Standard formExpanding this expression, we have ...
f(x) = x(x² -8x +41) +4(x² -8x +41) = x³ -8x² +41x +4x² -32x +164
f(x) = x³ -4x² +9x +164
Graph
The attached graph verifies the real zero (x=-4) and the value at x=1. It also shows that the factor with complex roots has vertex form (x -4)² +25, exactly as it should be.
Answer:
[tex]f(x) = (x+4)(x^2-8x+41)[/tex]
Step-by-step explanation:
Ok, so there are a couple of things to note here. The first thing is that there is a complex solution
Complex Conjugate Root Theorem:
if [tex]a-bi[/tex] is a solution then [tex]a+bi[/tex] is a solution and vice versa
Fundamental Theorem Of Algebra:
Any polynomial with a degree "n", will have "n" solutions. Those solutions can be real and imaginary numbers
So since we're given the root: [tex]4+5i[/tex], we can use the Complex Conjugate Root Theorem to assert that: [tex]4-5i[/tex] is also a solution.
So now we know 3 solutions/zeroes, and since n=3 (the degree), we can know for a fact that we have all the solutions due to the Fundamental Theorem of Algebra.
So using these roots, we can express the polynomial as it's factors. When you express a polynomial as factors it'll look something like so: [tex]f(x) = a(x-b)(x-c)(x-d)...[/tex] where a, b, and d are zeroes of the polynomial. Also notice the "a" value? This will affect the stretch/compression of the polynomial.
So let's express the polynomial in factored form:
[tex]f(x) = a(x-(-4))(x-(4+5i))(x-(4-5i))[/tex]
Simplify the x-(-4)
[tex]f(x) = a(x+4)(x-(4+5i))(x-(4-5i))[/tex]
Now let's distribute the negative sign to the complex roots
[tex]f(x) = a(x+4)(x-4-5i)(x-4+5i))[/tex]
Now let's rewrite the two factors (x-4-5i) and (x-4+5i) so the (x-4) is grouped together
[tex]f(x) = a(x+4)((x-4)-5i)((x-4)+5i))[/tex]
If you look at the two complex factors, this looks very similar to the difference of squares: [tex](a-b)(a+b) = a^2-b^2[/tex]
In this case a=(x-4) and b=5i. So let's use this identity to rewrite the two factors
[tex]f(x) = a(x+4)((x-4)^2-(5i)^2)[/tex]
Let's expand out the (x-4)^2
[tex]f(x) = a(x+4)(x^2+2(-4)(x)+(-4)^2-(5i)^2)[/tex]
Simplify
[tex]f(x) = a(x+4)(x^2-8x+16-(5i)^2)[/tex]
Now simplify the (5i)^2 = 5^2 * i^2
[tex]f(x) = a(x+4)(x^2-8x+16-(-25))[/tex]
Simplify the subtraction (cancels out to addition)
[tex]f(x) = a(x+4)(x^2-8x+41)[/tex]
So just to check for the value of "a", we can substitute 1 as x, and set the equation equal to 170
[tex]170 = a(1+4)(1^2-8(1)+41)\\170 = a(5)(1-8+41)\\170 = a(5)(34)\\170 = 170a\\a=1[/tex]
In this case it's just 1, so the polynomial can just be expressed as:
[tex]f(x) = (x+4)(x^2-8x+41)[/tex]
Related Questions
3. Complete the table (showing work) and draw a graph of the logarithmic function f(x) = log 1/5 x
Answers
Answer:
Step-by-step explanation:
Nereo and Azeneth start recording video at the same time. Nereo's phone starts with 1,000 megabytes
(MB) of free space and uses 7 MB of space per second of video.
Azeneth's phone has 1,255 MB of free space and uses 10 MB of space per second of video.
How much free space will they have when they first have the same amount of space left?
Answers
Nereo: y = -7x + 1000
Azeneth: y = -10x + 1255
Set equal and solve
-7x + 1000 = -10x + 1255
3x + 1000 = 1255
3x = 255
x = 85 seconds
Plug back in to find space left
-7(85) + 1000 = -10(85) + 1255
-595 + 1000 = -850 + 1255
405 = 405 MB
In how many possible ways can this be accomplished if there are 25 members on the board of directors
Answers
Explanation:
There are 25 choices for president. Then we have 25-1 = 24 choices for the secretary, and 24-1 = 23 choices for treasurer. This countdown (25,24,23) is because any given person cannot serve in more than one position.
Multiply out those values: 25*24*23 = 13800
Side note: you could use the nPr permutation formula as an alternative path. Plug in n = 25 and r = 3.
Identify the slope and y-intercept of the line y = ×/2 -7
Answers
Answer:
Slope: [tex]-\frac{1}{5}[/tex] Y-intercept: (0,0)
Step-by-step explanation:
By putting this into a graph you can see that the y-intercept is (0,0). Also, by looking at the chart and using the walk, rise rule you can see that the slope is [tex]-\frac{1}{5}[/tex].
Find the surface area of the composite figure 4 cm 13 cm 3 cm 4 cm 12 cm SA = [?] cm² 5 cm 3 cm.
Answers
12x4=48
side of rectangle ( 2 of them)
4x3x2=24
length of rectangle (2 of them)
12x3x2=72
area of triangle facing us ( 2 of them)
1/2x5x12x2= 60
top part of triangle
13x4= 52
back part of triangle :
4x5=20
total = add them all together
= 276
Find the value of z.
answer choices
A. 40
B. 56
C. 62
D. 80
Answers
Answer:
40°
Step-by-step explanation:
Using external angle intercepted arcs calculation:
50° = 1/2 (210-x) shows x = 110°
then z = 360 - 210 -110 = 40°
please help, I have been on this question for a while. Lotta yummy PTS
Answers
Answer:
[tex]y=\left|\dfrac{1}{2}x-\dfrac{5}{2}\right|+3[/tex]
Step-by-step explanation:
Given equation:
[tex]y=\left|\dfrac{1}{2}x-2\right|+3[/tex]
Translation
[tex]f(x-a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units right}[/tex]
When translating a graph to the right, subtract the number of units from the x variable.
Translating the given graph one unit to the right:
[tex]f(x-1) \implies y=\left|\dfrac{1}{2}(x-1)-2\right|+3[/tex]
Simplifying:
[tex]\implies y=\left|\dfrac{1}{2}(x-1)-2\right|+3[/tex]
[tex]\implies y=\left|\dfrac{1}{2}x-\dfrac{1}{2}-2\right|+3[/tex]
[tex]\implies y=\left|\dfrac{1}{2}x-\dfrac{5}{2}\right|+3[/tex]
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PLS HELP ME ASAP I NEED THE ANSWER NOW BEFORE 5 mins PLS Help ME I WILL MARK BRAINLIST IF YOUR ANSWER IS CORRECT AND WAS ANSWERED 4 MIN AGO HELP ME PLSSS
The table below shows that 1 day equals 24 hours. Notice that 12 hours is one-half of that. Use the table to help you determine how many days are equal to 42 hours.
1 day
24 hours
12 hours 12 hours
1 and 1 over 4. days (wrong)
1 and 1 over 2. days
1 and 3 over 4. days
2 days
Answers
Answer:
42 - 24 = 18
18/24 = 3/4
the answer would be 1 and 3/4 days
Answer:
1 and 3 over 4 days
Step-by-step explanation:
Since 24 hours is 1 day all we have to do is divide 42 into 24 and see what the resulting fraction is
42/24 = 7/4 (reducing to lowest fraction)
7/4 = 1 3/4 days (7 ÷ 4 gives quotient 1 and remainder is 3)
Answer:
1 and 3 over 4 days
Find the value of x
Answers
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
x = 12°[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
[tex]\qquad❖ \: \sf \:2x + 6 + 5x + 90 = 180[/tex]
[ by linear pair ]
[tex]\qquad❖ \: \sf \:7x + 6 = 180 - 90[/tex]
[tex]\qquad❖ \: \sf \:7x + 6 = 90[/tex]
[tex]\qquad❖ \: \sf \:7x = 90 - 6[/tex]
[tex]\qquad❖ \: \sf \:7x = 84[/tex]
[tex]\qquad❖ \: \sf \:x = 84 \div 7[/tex]
[tex]\qquad❖ \: \sf \:x = 12[/tex]
[tex] \qquad \large \sf {Conclusion} : [/tex]
[tex]\qquad❖ \: \sf \:x = 12 \degree[/tex]
Will mark brainliest
Answers
The plot of [tex]\sqrt{25-x^2}[/tex] is that of the top half of a circle with radius 5. The interval [-5, 0] captures the left half of this semicircle.
Adding 5 to this shifts the plot up by 5 units. The area under this curve is then the combined area of a square with side length 5 and the area of a quarter circle with radius 5.
So we have
[tex]\displaystyle \int_{-5}^0 5 + \sqrt{25-x^2} \, dx = 5^2 + \frac\pi4\cdot5^2 = \boxed{25 + \frac{25\pi}4}[/tex]
Two triangles are similar. The sides of the first triangle are 4,5,and 6. The largest side of the second triangle is 24. Find the perimeter of the second triangle.
Answers
Answer:
The perimeter of the second triangle is 60 units.Step-by-step explanation:
If two triangles are similar, their corresponding sides will be k times greater or smaller, where k is the scale factor.
We have two corresponding sides given, 6 and 24 or 6k. Using this info, find the value of k:
k = 24/6 = 4The other sides of the larger triangle are:
4k = 4*4 = 165k = 5*4 = 20The perimeter is the sum of sides:
P = 16 + 20 + 24 = 60can I return notebooks to target?
Answers
Answer: yes, as long as you haven't used them. :)
Step-by-step explanation:
BRIANLY FOR THE BEST EXPLANATION AND ANSWER!
Question: A circle has an area of 16mm². Find it's circumference.
Answers
Answer:
C ≈ 14.18
Step-by-step explanation:
Question Given:
A circle has an area of 16mm². Find it's circumference.
Formula:
C = 2√πA
Solve:
C = 2√πA
C = 2√(3.14 × 16)
C = 2√50.24
C = 14.1760361173
C ≈ 14.18
Hence, Circumference is approximately 14.18.
~Kavinsky
[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Here we go ~
Area of Circle :
[tex]\qquad \sf \dashrightarrow \: \pi {r}^{2} [/tex]
Now, it's given that area of circle is : 16 mm²
Let's calculate Radius ~
[tex]\qquad \sf \dashrightarrow \: \pi {r}^{2} = 16[/tex]
[tex] \qquad \sf \dashrightarrow \: {r}^{2} = \cfrac{16}{ \pi} [/tex]
[tex]\qquad \sf \dashrightarrow \: r = \sqrt{ \cfrac{16}{ \pi} } [/tex]
[tex]\qquad \sf \dashrightarrow \: r = { \cfrac{4}{ \sqrt{ \pi} } }[/tex]
Now,
[tex]\qquad \sf \dashrightarrow \: circumference = 2\pi r[/tex]
[tex]\qquad \sf \dashrightarrow \: c = 2 \pi \cfrac{4}{ \sqrt{ \pi} } [/tex]
[tex]\qquad \sf \dashrightarrow \: c = 8 \sqrt{ \pi} [/tex]
[ for approximate value take pi = 3.14 ]
[tex]\qquad \sf \dashrightarrow \: c = 8 \sqrt{3.14} [/tex]
[tex]\qquad \sf \dashrightarrow \: c \approx 8 \times 1.77[/tex]
[tex]\qquad \sf \dashrightarrow \: c \approx 14.18 \: \: mm[/tex]
Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers.
Find the equation of the circle whose center and radius are given.
center ( 5, 6), radius = 3
Answers
The equation of the circle whose center and radius are given as center ( 5, 6), radius = 3 is; x² + y² - 10x - 12y + 52 = 0
How to find the equation of a circle?We know that the general form of equation of a circle whose center is (h, k) and radius is r is expressed as;
(x - h)² + (y - k)² = r² --- (1)
We are given;
Center coordinate; (h, k) = (5, 6)
radius; r = 3
Plug in the values of h = 5 , k = 6 and r = 3 into the equation to get;
(x - 5)² + (y - 6)² = 3²
Expanding the equation gives;
x² - 10x + 25 + y² - 12y + 36 = 9
x² + y2 - 10x - 12y + 52 = 0
Thus the equation of the circle whose center and radius are given as center ( 5, 6), radius = 3 is; x² + y² - 10x - 12y + 52 = 0
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if the diameter of a circular pond is x meter what is the measure of its circumference
Answers
πx
Step-by-step explanation:
well pi is defined as the ratio of a circle's circumference to its diameter so just πx
Simplify Negative 3 over 2 divided by 9 over 6. −4 −1 4 1
Answers
Answer:
Step-by-step explanation: -3/2 divided by 9/6
= -3/2 multiply (to divide fractions we multiply by their recipical) 6/9
Then, simplify
= -3/3 = 1
Answer: -1
Hope this helps lol
The simplified result is -1. The correct option is b.
Given that:
Expression, -3/2 ÷ 9/6
To solve the given expression, follow as:
Rewrite the division as multiplication by the reciprocal of the second fraction.
-3/2 ÷ 9/6 = -3/2 × (6/9)
Now, simplify the fractions:
-3/2 × (6/9) = -3/2 × (2/3)
-3/2 × (6/9) = - (3 × 2)/(2 × 3)
-3/2 × (6/9) = -6/6
Finally, the simplified expression is -6/6. However, further simplify it by dividing the numerator and denominator by their greatest common divisor, which is 6:
-6/6 = -1
So, the simplified result is -1. The correct option is b.
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Complete question:
Simplify Negative 3 over 2 divided by 9 over 6.
a. −4
b. −1
c. 4
d. 1
Between which two integers is the value of √11?
9 and 10
5 and 6
7 and 8
O3 and 4
1 and 2
Answers
The square root of 11 is ?
The square root of 16 is 4
Between 3 and 4
In a football tournament, each team played exactly 19 games. Teams get three points for everyone and one point for every time. At the end of the tournament, team Olympus got a total of 28 points. How many could team Olympus have tied?
Answers
We can actually deduce here that when the tournament ended, team Olympus have tied for 13, 10, 7, 4, 1 times.
What is inequality?In Mathematics, inequality is used to express or show quantities that are less than, greater than, greater than or equal to, less than or equal to.
The following can be seen in inequality expressions:
≤≥ ><≠Looking at the question, we can see that each team plays exactly 19 games.
Every win (w) = 3
Every tie (t)= 1
Every loose = 0
But Olympus got a total of 28 points
Thus: 3w + t = 28 ---------(1)
Note that each team plays 19 games
Thus, T + W ≤ 19 --------------(2)
and T , W ≥ 0
Looking at equation (1) and (2)
=> 2W ≥ 9
=> W ≥ 4.5
( match can be won in integers only )
=> W ≥ 5
W - 5, 6, 7, 8, 9
T - 13, 10, 7, 4, 1
Loose - 1, 3, 5, 7, 9
Thus, team Olympus have tied for 13, 10, 7, 4, 1 times.
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The Bourassas decide to sell a home for $440,000. They are charged a real estate commission of 8% of the selling price, title insurance that is 1.4% of the selling price, and an escrow fee of $775.
(a)
What amount (in dollars) do the Bourassas receive after fees?
$
(b)
What percentage of the selling price was fees? Round to the nearest tenth of a percent.
%
Answers
The amount received net of expenses is $397865 and the percentage of the sale price was 10%.
Given that Bourassas decides to sell a house for $ 440,000. You will be charged a real estate fee of 8% of the sale price, property insurance of 1.4% of the sale price, and an escrow fee of $ 775.
The declared retail price is $ 440,000.
Real estate commission = 8% of the sale price
Real Estate Fee = (8/100) × 440000
Real Estate Fee = 8 × 4400
Real Estate Fee = $ 35,200
Security insurance costs of the title = 1.4% of the sale price
Cost of title insurance = (1.4 / 100) × 440 000
Cost of title insurance = 1.4 × 4400
Title Insurance Cost = $ 6160
The escrow fee assigned is $ 775.
(a) In determining the amount received after deducting the sale price of the property commission, title insurance and escrow fees, we receive
Amount received after fees = sale price - real estate commission - title insurance - escrow fees
Amount received after fees = 440000-35200-6160-775
The amount received after fees = $397865
(b) To find the percentage of sale price, add together all the terms real estate commission, title insurance and escrow fees and divide by the sale price we get
Percentage of selling price=((35200+6160+775)/440000)×100
Percentage of sale price = (42135/440000) × 100
Percentage of sale price = 9.57
Percentage of sale price = 10% (rounded down)
Hence Bourassas decides to sell a house for $440,000, then the amount Bourassas will receive after expenses is $397,865 and the percentage of the sale price of expenses is 10%.
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Complete the table of inputs and outputs for the function.
f(x) = -5(x + 7)
X
-9
01
0
f(x)
0
-60
Answers
Answer:
10, -7, -35, 5
Step-by-step explanation:
f(9) = -5(-9 + 7) = 10
0 = -5(x + 7)
0 = x + 7
x = -7
f(0) = -5(0 + 7) = -35
-60 = -5(x + 7)
12 = x + 7
x = 5
solve this equations 3x+4x-8=6(x-3)+x
Answers
Answer:
no solutions
Step-by-step explanation:
5
1
The measure of Z1 =
O 55
O62.5
110
5
Answers
Answer: 55
Step-by-step explanation:
Congruent angles intercept congruent chords, so since the circumference of a circle is 360 degrees, the arc angle 1 is inscribed in measures 110 degrees.
So, by the inscribed angle theorem, angle 1 measures 55 degrees.
Congruent angles intercept congruent chords, so since the circumference of a circle exists 360 degrees, arc angle 1 exists inscribed in measures 110 degrees.
So, by the inscribed angle theorem, angle 1 measures 55 degrees.
What are congruent angles?Congruent angles exist as two or more angles that exist identical to each other. Therefore, the measure of these angles exists equivalent to each other. The kind of angles does not create any distinction in the congruence of angles, which implies they can be acute, obtuse, exterior, or interior angles.
The Inscribed Angle Theorem states that the measure of an inscribed angle exists half the measure of its intercepted arc. Inscribed angles that intercept the exact arc exist congruent. This exists named the Congruent Inscribed Angles Theorem and exists shown below.
∠ A D B and ∠ A C B intercept, so m ∠ A D B = m ∠ A C B.
Congruent angles intercept congruent chords, so since the circumference of a circle exists 360 degrees, arc angle 1 exists inscribed in measures 110 degrees.
So, by the inscribed angle theorem, angle 1 measures 55 degrees.
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Three times a certain number decreased by 5 is equal to 31.
Answers
hope this helps!
If the r-value, or correlation coefficient, of a data set is 0.854, what is the
coefficient of determination to three decimal places?
A. 0.829
• B. 0.754
• C. 0.729
D. 0.854
Answers
Coefficient of determination(r²), is defined as, how well the data values are associated with each other when Regression line is drawn. In regression analysis, the coefficient of determination measures , how well the regression predictions approximate the real data points, means it measures the closeness between two variables.The Value of r², lies between 0 to 1. If value of r²=1, it shows ,regression line that is data values are Perfectly associated with each other.
If ,r²=0, it means there is no variation between two variables.There is 0% variation between two variables.
→Coefficient of Variation=[Correlation coefficient]²
=r²
=(0.854)²
=0.854 × 0.854
=0.729316
=0.730(Approx)
If AC is a diameter and Arc AD = 90. Find ∠DAC. Round your answer to the nearest tenth.
Answers
Answer:
45°
Step-by-step explanation:
it is an inverted angle within a circle
==========================================================
Explanation:
Minor arc AD is the shortest path from A to D along the circle's edge. This is 90 degrees. Minor arc AD combines with DC to get arc ADC
Arc ADC is a semicircle because of the diameter AC. Any semicircle has a measure of 180 degrees.
So,
(minor arc AD) + (minor arc DC) = arc ADC
(minor arc AD) + (minor arc DC) = 180
(90) + (minor arc DC) = 180
minor arc DC = 180 - 90
minor arc DC = 90
AD and DC are 90 degrees each.
Then notice that inscribed angle DAC subtends minor arc DC. Use the inscribed angle theorem to determine angle DAC is 90/2 = 45 degrees
Which of the graphs below shows the solution set of -7 < x <0
Answers
Answer:
none
Step-by-step explanation:
A. -7 <= x < 0
B. x <= -7 or x > 0
C. -7 < x <= 0
D. x < -7 or x >= 0
To create a graph to represent -7 < x < 0, draw empty circles at -7 and 0, then connect the two with a line.
Please help with this geometry question!
Answers
Answer:
second option
Step-by-step explanation:
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
Correct choice is 2Since the two triangles ABE and ADE are congruent, their corresponding sides and angles are equal.
After observing all the options, we can conclude that :
Only angle AED and angle AEB are angles of given triangles and are actually equal.
[tex] \qquad \large \sf {Conclusion} : [/tex]
Option 2 is correctwhich of the following rational functions is graphed below?
Answers
Answer:
c
Step-by-step explanation:
Oblique Asymptotes
R(x) will have oblique asymptote if it can be represented in the form
1 / (Q)x
A man can take a same time to row 13km downstream and 7km upstream.His speed in still water is 5km/hr.The speed of stream is
Answers
Answer:
0.6 km/hr
Step-by-step explanation:
Speed downstream: 13/5= 2.6 km/hr
Speed upstream: 7/5= 1.4 km/hr
Therefore, Velocity of current is: 1/2 (2.6-1.4) km/hr = 1/2(1.2) km/hr
Evaluate f(x)=−4ex−2−4 for x=4. round to the nearest 4 decimal
Answers
Answer:
25.2
Step-by-step explanation:
I'll assume you wrote:
[tex]f(x) = 4e^{x-2} - 4[/tex]
So when x = 4:
[tex]4e^{4-2} - 4 = 4e^2 - 4[/tex] ≅ [tex]4\cdot 7.3 - 4 = 25.2[/tex]
