Answers
As there is the highest degree 4 there are four zeros .
Using calculator
the roots are(attached)
The graph is also attached
Answer:
a) 4 roots
b) x = 1 and x = -3
[tex]\textsf{c)} \quad x=1, \quad x=-3, \quad x=\dfrac{-1 + \sqrt{3}i}{2}, \quad x=\dfrac{-1 - \sqrt{3}i}{2}[/tex]
d) see attached
Step-by-step explanation:
Given function:
[tex]f(x)=x^4+3x^3-x-3[/tex]
Part (a)
From inspection of the function, the highest exponent of the polynomial is 4. Therefore, it is expected that the polynomial will have 4 roots.
Part (b)
To find the rational roots, find values of x where [tex]f(x)=0[/tex]
[tex]f(1)=(1)^4+3(1)^3-1-3=0[/tex]
[tex]f(-3)=(-3)^4+3(-3)^3-(-3)-3=0[/tex]
Therefore, the rational roots are: x = 1 and x = -3
Part (c)
As there are only 2 rational roots, the other 2 roots must be complex roots. To find these, first factor the polynomial using the 2 rational roots found in part (b):
[tex]\implies f(x)=(x-1)(x+3)(x^2+bx+1)[/tex]
[tex]\implies f(x)=(x^2+2x-3)(x^2+bx+1)[/tex]
[tex]\implies f(x)=x^4+bx^3+x^2+2x^3+2bx^2+2x-3x^2-3bx-3[/tex]
[tex]\implies f(x)=x^4+(2+b)x^3+(2b-2)x^2+(2-3b)-3[/tex]
Compare coefficients to find b:
[tex]\implies (2+b)x^3=3x^3[/tex]
[tex]\implies b=1[/tex]
Therefore the fully factored function is:
[tex]\implies f\:\!(x)=(x-1)(x+3)(x^2+x+1)[/tex]
Therefore:
[tex](x^2+x+1)=0[/tex]
To find the complex roots, use the quadratic formula
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{1^2-4(1)(1)}}{2(1)}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{-3}}{2}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{3}\sqrt{-1}}{2}[/tex]
[tex]\implies x=\dfrac{-1 \pm \sqrt{3}i}{2}[/tex]
Therefore, all the roots of the function are:
[tex]x=1, \quad x=-3, \quad x=\dfrac{-1 + \sqrt{3}i}{2}, \quad x=\dfrac{-1 - \sqrt{3}i}{2}[/tex]
Part (d)
End behaviors:
As the degree of the function is even and the leading coefficient is positive, the end behaviors are:
[tex]\textsf{As } x \rightarrow - \infty, \:\: f(x) \rightarrow + \infty[/tex]
[tex]\textsf{As } x \rightarrow + \infty, \:\: f(x) \rightarrow + \infty[/tex]
y-intercept
Substitute x = 0 into the function:
[tex]f\:\!(0)=(0)^4+3(0)^3-(0)-3=-3[/tex]
Therefore, the y-intercept is (0, -3)
To find the approximate x-coordinates of the turning points (stationary points), differentiate the function, set it to zero, then solve for x:
[tex]f'\:(x)=4x^3+9x^2-1[/tex]
[tex]\implies 4x^3+9x^2-1=0[/tex]
Using a calculator, x ≈ -2.2, x ≈ -0.4, x ≈ 0.3
Therefore the approximate coordinates of the turning points are:
(-2.2, -9.3), (-0.4, -2.7) and (0.3, -3.2)
We don't need to plot these points, they merely help us with the approximate shape of the curve.
Related Questions
What is the probability that the manufacturing unit has carbon emission beyond the permissible emission level and the test predicts this? a. 0.2975 b. 0.0525 c. 0.0975 d. 0.5525 e. 0.6325
Answers
The conditional probability that the carbon emission is beyond the permissible emission level and the test predicts this is given by:
a. 0.2975.
What is Conditional Probability?Conditional probability is the probability of one event happening, considering a previous event. The formula is:
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which:
P(B|A) is the probability of event B happening, given that A happened.[tex]P(A \cap B)[/tex] is the probability of both A and B happening.P(A) is the probability of A happening.In this problem, we have that the events are as follows:
Event A: Carbon emission beyond the permissible emission level.Event B: Test predicts this.We have that 35% of the units have carbon emission beyond the permissible emission level, and the test is 85% accurate, hence:
[tex]P(A) = 0.35, P(B|A) = 0.85[/tex]
Then:
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
[tex]0.85 = \frac{P(A \cap B)}{0.35}[/tex]
[tex]P(A \cap B) = 0.85(0.35) = 0.2975[/tex]
Which means that option a is correct.
More can be learned about conditional probability at https://brainly.com/question/14398287
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Please help me with my math please?
Answers
Answer:
y = y coordinate
m = slope
x = x coordinate
b = y intercept
Line passes through (2, -1) and slope of 1/3
Step 1y = mx + b, Identify: x = 2, y = -1, slope: [tex]\sf \frac{1}{3}[/tex]
insert following values
[tex]\rightarrow \sf -1 = \dfrac{1}{3} (2) + b[/tex]
multiply
[tex]\rightarrow \sf -1 = \dfrac{2}{3} + b[/tex]
switch sides
[tex]\rightarrow \sf b = -1 - \dfrac{2}{3}[/tex]
subtract
[tex]\sf \rightarrow b = -\dfrac{5}{3}[/tex]
convert to mixed fraction
[tex]\sf \rightarrow b = -1\dfrac{2}{3}[/tex]
Step 2y = mx + b, m = [tex]\sf \frac{1}{3}[/tex], b = [tex]-1\frac{2}{3}[/tex]
insert following values
[tex]\rightarrow \sf y = \dfrac{1}{3} x -1\dfrac{2}{3}[/tex]
Question Write an equation in slope-intercept form of the line that passes through (6, −2) and (12, 1)
Answers
Answer:
[tex]y = \dfrac 12x -5[/tex]
Step-by-step explanation:
[tex]\text{Given that,}~ (x_1,y_1) = (6,-2)~ \text{and}~ (x_2,y_2) = (12,1)\\\\\text{Slope,}~ m = \dfrac{y_2 -y_1}{x_2 -x_1} = \dfrac{1+2}{12-6}= \dfrac{3}{6} = \dfrac 12\\\\\text{Equation of line,}\\\\~~~~~~~~y-y_1 =m(x-x_1)\\\\\implies y+2=\dfrac 12 (x-6)~~~~~~~~~~~~~~;[\text{Point -slope form}]\\\\\implies y = \dfrac 12 x -3 -2\\\\\implies y = \dfrac 12x -5~~~~~~~~~~~~~~~~~~~~~;[\text{Slope-intercept form.}][/tex]
what is the area of a triangular prism?
Answers
Answer:
14! i think :D
Step-by-step explanation:
im probly worngg TvT
Need help please answer ASAP!
(20-21)
Answers
#20
1/2x+3<2x-61/2x-2x<-6-3-3/2x<-93/2x>9x>9(2/3)x>6#21
Find LCD of 2,7x,x
x is most commonSo
LCD is 2(7x)(x)=14x²Answer:
20. (4) x > 6
21. (3) 14x
Step-by-step explanation:
Question 20
[tex]\begin{aligned}\dfrac{1}{2}x+3 & < 2x-6\\\dfrac{1}{2}x+3-3 & < 2x-6-3\\\dfrac{1}{2}x & < 2x-9\\\dfrac{1}{2}x-2x & < 2x-9-2x\\-\dfrac{3}{2}x & < -9\\-\dfrac{3}{2}x \cdot 2 & < -9 \cdot 2\\-3x & < -18\\-\dfrac{3x}{3} & < -\dfrac{18}{3}\\-x & < -6\\\dfrac{-x}{-1} & < \dfrac{-6}{-1}\\x & > 6\end{aligned}[/tex]
Question 21
Rewrite all three fractions so that their denominators are the same:
[tex]\dfrac{1}{2}=\dfrac{1 \times 7x}{2 \times 7x}=\dfrac{7x}{14x}[/tex]
[tex]\dfrac{2}{7x}=\dfrac{2 \times 2}{7x \times 2}=\dfrac{4}{14x}[/tex]
[tex]\dfrac{5}{x}=\dfrac{5 \times 14}{x \times 14}=\dfrac{70}{14x}[/tex]
Therefore, the least common denominator is [tex]14x[/tex]
Question 39 of 40
The circle below has a radius of 8 centimeters. What is the area of the shaded
region?
X
120°
8 cm
OA. 32 square centimeters
B. 8 square centimeters
64
3
square centimeters
square centimeters
O C.
O D.
16
3
Answers
Convert angle to radians
120=2π/3So
Area
r²Ø/28²(2π/3)(1/2)64π/3cm²Answer:
[tex]\textsf{C.} \quad \dfrac{64 \pi}{3} \: \sf square\:centimeters[/tex]
Step-by-step explanation:
Formula
[tex]\textsf{Area of a sector of a circle}=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2[/tex]
[tex]\textsf{(where r is the radius and the angle }\theta \textsf{ is measured in degrees)}[/tex]
Given:
[tex]\theta[/tex] = 120°r = 8 cmSubstitute the given values into the formula and solve for Area:
[tex]\large \begin{aligned}\implies \textsf{Area} & =\left(\dfrac{120^{\circ}}{360^{\circ}}\right) \pi (8)^2\\\\& =\left(\dfrac{1}{3}\right) \pi (64)\\\\& =\dfrac{64}{3} \pi \: \sf cm^2\end{aligned}[/tex]
What is the solution to this system of equations? 2x + 2y = 8 and 4x + 3y = 16
Answers
Answer:
y=0, x=4
Step-by-step explanation:
2x+2y=8 --- (1)
4x+3y=16 --- (2)
Multiply the coefficient of x in equation (1) across the variables in equation (2), and multiply the coefficients of x in equation (2) across the variables in equation (1).
8x+8y=32 ---- (3)
8x+6y=32 ---- (4)
Subtract equation (3) from (4)
2y=0, y=0/2, y=0.
Substitute the solution for y=0 into equation (1)
2x+2(0)=8
2x=8
x=4
Book-club members are required to buy a minimum number of books each year. Leslee bought 3 times the minimum. Denise bought 7 more than the minimum. Together, they bought 23 books. What is the minimum number of books?
Answers
Answer: 4 books
Step-by-step explanation:
x = minimum number of book
Leslee
= 3x (bought 3 times the minimum)
Denise
= x + 7 (bought 7 more than the minimum)
Together
= (3x) + (x + 7)
(3x) + (x + 7) = 23 (books they bought together)
3x + x + 7 = 23
3x + x = 23 - 7 (isolate the like variables together)
4x = 16 (divide both sides by 4)
x = 4
4 is the minimum number of books
To check:
3(4) + (4+7)
12 + 11 = 23
I need help on this question
Answers
the opposites would equal each other.
The last 2 answers are the correct ones.
HELP PLEASE ILL GIVE BRAINLEST
Answers
Answer:
y= 40 degree
Step-by-step explanation:
ACD+CDA+DAC= 180 degree
y+ 90+30+20= 180 degree
y+140 = 180
y= 180- 140 degree
y= 40 degree
Hope it helps you :)))))
have a good day
help with practice problem
Answers
Answer:
i think the answer is r=0.5 or r=−3/2−x^2
Step-by-step explanation:
-4r^2-4r=6
step 1. -4r(-4r)= 16r
16r-4r=6
step 2. 16r-4r=12r
12r=6
step 3. /12 on both sides
r= 0.5
A container is shaped like a triangular prism. The height of the container is 15 centimeters, and the volume of the container is 180 cubic centimeters. What is the area of the base of the container in square centimeters?
a. 10 cm2
b. 12 cm2
c. 18 cm2
d. 20 cm2
Answers
The volume of this rectangular prism is 48,608 cubic centimeters. What is the value of b?
Answers
Answer:
b = 49 cm
Step-by-step explanation:
Volume of a rectangular prism = width × length × height
Given:
volume = 48,608 cm³width = 16 cmlength = bheight = 62 cmSubstituting the given values into the formula and solving for b:
⇒ 48608 = 16 × b × 62
⇒ 48608 = 992 b
⇒ b = 48608 ÷ 992
⇒ b = 49
Answer:
b = 49 cm
Step-by-step explanation:
Volume of a rectangular prism: length × width × height
V = L×W×H
v = 48,608 cm³l = b cmw = 16 cmh = 62 cmSubstitute the values above and solve for b:
48,608 = b × 16 × 62
48,608 = 992b
48,608/992 = 992b/992
49 = b
Final answer: 49 cm
Hope this helps!
2. A sphere has a great circle whose circumference is 95 centimeters. Which of the following is closest to the
volume of the sphere, in cubic centimeters?
1) 4,600
(3) 12,700
2) 7,500
(4) 14,500
Answers
Answer:
Approximately [tex]14,\!500\; {\rm cm^{3}}[/tex].
Step-by-step explanation:
The radius of a sphere is the same as the radius of the great circle of that sphere.
Thus, in this question, it would be possible to find the radius of the sphere by finding the radius of the great circle.
The circumference of a circle of radius [tex]r[/tex] is [tex]2\, \pi \, r[/tex]. In this question, it is given that the circumference of this great circle is [tex]95\; {\rm cm}[/tex]. Thus, the radius of this great circle would be:
[tex]\begin{aligned}r &= \frac{95\; {\rm cm}}{2\, \pi} \approx 15.1197\; {\rm cm}\end{aligned}[/tex].
Thus, radius of the sphere in this question would also be approximately [tex]15.1197\; {\rm cm}[/tex].
The volume of a sphere of radius [tex]r[/tex] is [tex](4/3)\, \pi\, r^{3}[/tex]. Thus, the volume of this sphere of radius [tex]r = 15.1197\; {\rm cm}[/tex] would be approximately:
[tex]\begin{aligned}\frac{4}{3} \, \pi\times (15.1197\; {\rm cm})^{3} \approx 14,\!500 \; {\rm cm^{3}}\end{aligned}[/tex]
A cube with a side length of 6 feet is sliced diagonally such that it passes through the three vertices shown in the figure. which describes the cross section? a cube. a cross section is a triangle that touches 3 corners of the cube. a triangle with 3 equal sides measuring 6 feet a triangle with 3 equal sides measuring greater than 6 feet a triangle with only 2 equal sides measuring 6 feet a triangle with only 2 equal sides measuring greater than 6 feet
Answers
Answer:
Its C.
Step-by-step explanation:
Answer:
"C"
Step-by-step explanation:
It's "C" on the other brainly as well -_-
In a company's first year in operation, it made an annual profit of $247,500. The
profit of the company increased at a constant 16% per year each year. How much
total profit would the company make over the course of its first 10 years of operation,
to the nearest whole number?
Answers
To calculate this sort of problem:
⇒ involves calculating the total profits over each year
⇒ with a constant increase periodically
⇒ need to use the Compound Interest Equation
[tex]A = P(1+\frac{r}{n})^{nt}[/tex]
A: total amount after t yearsP: initial amount in accountr: rate per periodn: number of times the profit increases per yeart: number of yearsLet's fill in some variables:
P: $247500r: 16% increase (in equation form, we plugin 0.16)n: 1 time per yeart: 10 yearsWhat do we want to know:
A: total amount after 10 yearsLet's solve:
[tex]A= 247500*(1+\frac{0.16}{1} )^{1*10}=\\A=247500*(1+0.16)^{10}\\A=247500*(1.16)^{10} \\A= 1,091,830.18[/tex]
To the nearest whole number
Answer: $1,091,830Hope that helps!
I was confused on how to write the quadratic in standard form
Answers
Answer:
3x² + 4x + 8 = 0
Step-by-step explanation:
Standard form of quadratic equation is
ax²+bx+c=0
standard form of quadratic equation 3x²+4x+7=-1 is
3x² + 4x + 7 = -1
3x² + 4x + 7 + 1 = 0
3x² + 4x + 8 = 0
BRAINLIEST For the fastest answer
Hello I need help on this math question
Answers
Answer:
B
Step-by-step explanation:
B: $303.75
I hope this helps you and I hope you do well.
Find the center and radius of the circle represented by the equation below.
x² + y²- 6x - 12y +29=0
Answers
Answer:
radius 4
center (3,6)
Step-by-step explanation:
(x - h)^2 + (y - k)^2 = r^2
coordinates of the center (h, k) and the radius is (r)
x² + y²- 6x - 12y +29=0
x² - 6x + y²- 12y +29=0
(x² - 6x) + (y²- 12y) +29=0
complete the square
(x² - 6x) + 9 + (y²- 12y) +36 +29= + 9 +36
(x² - 6x + 9) + (y²- 12y + 36) = + 9 +36 -29
(x-3)^2 + (y-6)^2 = 16
(x - h)^2 + (y - k)^2 = r^2
coordinates of the center (h, k) and the radius is (r)
center (3,6)
radius 4
cuemathcom
varsitytutors
Answer: Center: (3,2) Radius: 5
Step-by-step explanation:#x^2 - 6x +y^2 - 4y = 12
Then take 1/2 of the 'b' term for both quadratic expressions, square those values and add them to both sides.
#x^2 -6x + 9 + y^2 - 4y + 4 = 12 + 9 + 4
(x - 3)^2 + (y -2)^2 = 25
Circle centered at (3,2) with radius = 5
Equation of a line that goes through the following points: (4,1) and (7,10)
Answers
[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{7}~,~\stackrel{y_2}{10}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{10}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{7}-\underset{x_1}{4}}}\implies \cfrac{9}{3}\implies 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{1}=\stackrel{m}{3}(x-\stackrel{x_1}{4}) \\\\\\ y-1=3x-12\implies y=3x-11[/tex]
Please Help I Don't Understand!
Answers
Answer:
the aa similarity postulate
Step-by-step explanation:
no sides are given and we can not simply assume the sides are congruent, however we do know two of the angles on both triangles are the same allowing us to know that the third therefore must also be the same.
both traingles share angle MAN
and AMN and ABC are equal so therefore
angle ANM and ACB must also be the same
Using the points ( 6.4, 117) and (6.6, 120), what is the slope of the trend line?
a. -15
b. -12
c. 12
d. 15
Answers
Answer:
d 15.
Step-by-step explanation:
Slope = (difference in y values)/ (corresponding difference in x values)
So we have:
Slope = (120-117) / (6.6-6.4)
= 3 / 0.2
= 30/2 = 15.
PLEASE HELP. I need this done by the end of today. plpease help im beggin
Answers
Answer:
I only did 1-16 but here:
11. x² + y² = 4
12. (x+3)² + (y-3)² = 1
13. x² + (y-3)² = 16
14. (x-7)² + (y+2)² = 4
15. x² + (y+20)² = 100
16. (x+4)² + (y+6)² = 16
what is the probability tho computer will select a point in the shaded region?
Answers
I HAVE A CYLINDER AND cone WITH THE
Same Dimensions (RADIUS AND HEIGHT OF
BOTH ARE THE SAme). The volume OF THE
cone is 33 CUBIC INCHES. WHAT IS THE
Volume OF THE CYLINDER IN CUBIC
inches?
125 Sunny
Answers
Answer:
99 in³
Step-by-step explanation:
Let the radius and height of each be r and h in. respectively.
Write an equation for the volume of cone.
Volume of cone= ⅓πr²h
⅓πr²h= 33
Multiply both sides by 3:
πr²h= 99
Volume of cylinder= πr²h
We have found the value of πr²h previously, which is 99.
Thus, the volume of the cylinder is 99 in³.
Given f(x) and g(x) = k•f(x), use the graph to determine the value of k.
Answers
Step-by-step explanation:
the slope of f(x) = 1
for every increase of x by 1 also y increases by 1.
the slow of g(x) = -2
for every increase of x by one y decreases by 2.
so, we see, as
g(x) = k × f(x)
k = g(x) / f(x) = slope g / slope f = -2/1 = -2
why ? because both functions are linear functions (lines).
they look like
y = ax + b
with a being the slope of the line.
so, when multiplied by k, this is kax + kb, and we see k as factor already in the slope of the line itself.
Joe's Fabric Store purchased 6 bolts of fabric at $150.00 per bolt. Each bolt contained 50 m of fabric. If the fabric was retailed at $8.00 per metre, calculate the profit or loss.
Answers
reasoning:
the store purchased 300 m of fabric at $900 (6 x 50 = 300; 6 x 150 = 900).
the retail price for all that fabric would be $2400 ($8 x 300 m = $2400).
$2400 - $900 = $1500
Can someone help me with this question?
Answers
Answer:
5 possible outcomes
1. All 4 are heads,
2. All 4 are tails,
3. 3 are heads and 1 tails,
4. 2 are heads and 2 are tails,
5. 1 is tails and three are heads.
Step-by-step explanation:
On one side of a coin it’s heads, and on the other it’s tails.
there are 4 coins
(Coin side, coin side, coin side, coin side)
Possibilities;
(Heads, heads, heads, heads)
all four coins are heads
(Heads, tails, heads, heads)
three coins are heads and one tails
(Heads, tails, tails, heads)
two coins are heads, two coins are tails
(Heads, tails, tails, tails)
one coin is heads, the other three are tails
(Tails, tails, tails, tails)
all four coins are tails.
5 possible outcomes
The table shows the annual sales revenues of different types of vehicles made by four automobile manufacturers.
The number in the highlighted cell is ______ (options are: a. the number of convertibles sold by Pluto Cars b. the revenue from sales of convertibles by Pluto Cars c. the revenue from sales of sedans by Pluto Cars d. the total sales revenue of Pluto Cars). The relative frequency of this vehicle category compared with the total sales revenue of Pluto Cars is _______ (options are: a. 0.107 b. 0.225 c. 0.29 d. 0.33)
Answers
Using the relative frequency concept, it is found that:
The number in the highlighted cell is: b. the revenue from sales of convertibles by Pluto CarsThe relative frequency of this vehicle category compared with the total sales revenue of Pluto Cars is: b. 0.225.What is a relative frequency?A relative frequency is given by the number of desired outcomes divided by the number of total outcomes.
The number in the cell has bounds conversible and Pluto Cars, and the table represents the amount earned, that is, revenue, hence option b is correct.
The total revenue of Pluto Cars is of $80 million, with convertibles corresponding to $18 million, hence the relative frequency is given by:
r = 18/80 = 0.225.
More can be learned about relative frequency at https://brainly.com/question/14398287
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pls help asap The scatter diagram shows the grades and number of missing assignments for students in a grade 8 math class
Answers
Answer A
