-The polynomial function p(x) = x² + 4x³ - 7x² - 22x + 24 has known factors of (x + 4) and (x - 1).
a. Rewrite p(x) as the product of linear factors.
b. Draw a rough sketch of the graph of the function.
Answers
Answer:
p(x) = (x +4)(x +3)(x -1)(x -2)see the first attachment for a graphStep-by-step explanation:
Given p(x) = x⁴ + 4x³ - 7x² - 22x + 24 with known factors (x +4) and (x -1), you want the function written as a product of linear factors, and a sketch of the graph.
A graphing calculator can help with both parts of this. It can show you the remaining zeros are -3 and 2, so the remaining linear factors are (x +3) and (x -2). At the same time, it produces a graph of the function. This is shown in the first attachment.
a. RewriteSynthetic division is a convenient way to find the remaining factors of the polynomial. Dividing by (x+4) gives ...
p(x) = (x +4)(x^3 -7x +6) . . . . . . shown in the second attachment
And dividing the cubic by (x -1) gives ...
p(x) = (x +4)(x -1)(x^2 +x -6) . . . . . . shown in the second attachment
The quadratic will have linear terms with constants that sum to 1 and have a product of -6. These constants are 3 and -2.
The rewrite of p(x) is ...
p(x) = (x +4)(x +3)(x -1)(x -2)
b. Graph
The 4th degree polynomial has a positive leading coefficient, so is above the x-axis at both the left and right ends of the graph. The graph crosses the x-axis at x = -4, -3, 1, and 2.
We note these roots are symmetrical about x=-1, so there will be a maximum at that point. That maximum is p(-1) = (-1 +4)(-1 +3)(-1 -1)(-1 -2) = 3·2·(-2)(-3) = 36. The minimum values will be found approximately halfway between -4 and -3, and again between -1 and -2. Those minima will be approximately p(-3.5) = (-.5)(.5)(-4.5)(-5.5) ≈ -6.2
The y-intercept is +24, the constant in the polynomial.
With the zero crossings, line of symmetry, local maximum, approximate local minima, and the y-intercept, we can make a passable sketch of the graph.
A graph is seen in the first attachment.
Related Questions
Please help simple math 50 points
Answers
Answer:
2 pizzas
Step-by-step explanation:
5 7/8 - 4 1/16=
5 14/16 - 4 1/16 ==> make the denominators equal to each other so they can be easily subtracted in the next step
5 + 14/16 - (4 + 1/16)=
5 + 14/16 - 4 - 1/16 =
5 - 4 + 14/16 - 1/16 =
1 + 13/16 = 1 13/16 which is approximately 2 pizzas
What is the value of this expression.
Answers
here's your answer hope it helps
What’s the answer plss I really need help
Answers
Answer:
The answer is d) 22.4
Step-by-step explanation:
I attached a file below if you want to see the process!
(MULTIPLE CHOICE) Write an expression that would represent the perimeter of a rectangle with a width of 10n and a length of 4.
A) 10n + 4 + 10n + 4
B) 10n + 4
C) 4(20n) (this is multiplication)
D) 10 + 4 + 20n + 4
Answers
by the time vince left the library he had 7/8 of his new book left to read ge read 1/10 of his book on the way home and 2/5 of the book after dinner what fraction of the book does he have left to read
Answers
We have the next information
When he left the library he read 8/8-7/8= 1/8
then he read 1/10
after dinner, he read 2/5
First, we need to sum the fraction of the book he read
[tex]\frac{1}{8}+\frac{1}{10}+\frac{2}{5}[/tex]we need to have the same denominator in order to sum the fractions
[tex]\frac{5+4+16}{40}=\frac{25}{40}=\frac{5}{8}[/tex]the book complete represent 8/8 so we need to subtract 5/8 from the fraction of the complete book
[tex]\frac{8}{8}-\frac{5}{8}=\frac{3}{8}[/tex]the fraction of the book does he have left to read is 3/8
Suppose that R(x) is a polynomial of degree 13 whose coefficients are real numbers. also, suppose that R(x) has the following zeros. answer the following.edit: if possible please double check the answers just to be safe.
Answers
(a) Complex zeros of a polynomial come in pairs.
If a + bi is a zero of a polynomial then its conjugate a - bi is also a zero of the polynomial.
The given complex zeros of R(x) are 1 + 3i and -2i.
1 - 3i is the conjugate of 1 + 3i.
Hence, another zero of R(x) is 1 - 3i
b)
Since the polynomial R(x) is of order 13 then R(x) must have 13 zeros.
The given complex zeros of R(x) are 1 + 3i and -2i. We also know that the conjugates of 1 + 3i and -2i are zeros of R(x). Hence, R(x) has at least 4 complex roots
Hence, the maximum number of real zeros of R(x) is (13 -4).
The maximum number of real zeros of R(x) is 9
c) Let the maximum number of nonreal zeros (complex roots) be n
Complex roots come in pairs. Therefore, n must be even.
Hence, n ≤ 13 - 1 = 12
n ≤ 10
We have been given a real zero of R(x), 3 ( With the multiplicity of 4).
12 - 4 = 8
Therefore,
n ≤ 8.
Hence the maximum number of nonreal zeros of R(x) is 8
Dwayne is making a poster about friendship in his art class. He divides his poster into 12 equal parts. Dwayne writes the names of his friends all over each part. Dwayne uses 7 parts for his neighborhood friends and 5 parts for his friends from school. What fraction of the poster does Dwayne use for his friends from school?
Answers
The fraction of the poster does Dwayne use for his friends from school is 5/12.
What is the fraction of the poster that Dwayne uses for his friends from school?A fraction is a non-integer that has a numerator and a denominator. The numerator is the number that is above the line and the denominator is the number below the line. An example of a fraction is 5/12.
In this question, the numerator is the number of parts he gives his friends from school. The denominator is the total number of parts he divides the poster.
Fraction = parts he gave his friend from school / total number of parts
5 / 12.
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Please help me will give brainly Thanks
Answers
The required equation of line would be 2y = 5x - 8 which is shown in the given graph.
What is the slope of the line?The slope of a line is defined as the angle of the line.
According to the given figure,
We clearly see that the coordinates of the y-intercept would be (0, -4).
The value of the x-coordinate is 2 when y-coordinate is 1.
Here the line passes through the points (0, -4) and (2, 1).
Let the required line would be y - y₁ = (y₂ - y₁)/(x₂ -x₁ )[x₂ -x]
x₁ = 0, y₁ = -4
x₂ = 2, y₂ = 1
⇒ y - y₂ = (y₂ - y₁)/(x₂ -x₁ )[x -x₂]
⇒ y - 1 = (1 - (-4))/(2 - 0 )[x -2]
⇒ y - 1 = 5/2[x -2]
⇒ 2y - 2 = 5[x -2]
⇒ 2y = 5x - 10 + 2
⇒ 2y = 5x - 8
Therefore, the required equation of line would be 2y = 5x - 8 which is shown in the given graph.
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The National Research Council recommends keeping a 1 : 3 copper to zinc ratio in a horse’s diet. If the horse is getting 420 milligrams (mg) of zinc per day, how many milligrams of copper does the Council recommend the horse have?
Answers
If he National Research Council recommends keeping a 1 : 3 copper to zinc ratio in a horse’s diet. If the horse is getting 420 milligrams (mg) of zinc per day, The number of milligrams of copper that the Council recommend the horse have is : 140 milligrams.
How to find the milligrams of copper?Given data:
Copper to zinc ratio = 1 : 3
Zinc per day = 420 milligrams
Hence,
1 /3 = x / 420
x /420 = 1 / 3
Where x is the number of milligrams of copper
so,
x = 1 /3 × 430
x = 140 milligrams of copper
Therefore the milligrams of copper is 140 milligrams.
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In the figure below fg = 18 and gh= 19 find fh
Answers
Answer:
18*19 BECAUSE 19 is a prime number and g has to be 1 since if g was anything else like 2 it wouldn't be the same for gh
5. Let the graph of g be a translation 2 units up and 2 units right, followed by a reflection in the x-axis of the graph off(x) = -(x+3)^2 - 2. Write a rule for g.
Answers
Answer:
[tex]g(x)=-f(x-2)+2[/tex]Explanation:
Given the function f(x)
A translation of 2 units up = f(x)+2
Next, a translation of 2 units right = f(x-2)+2
When we reflect the result above in the x-axis, we have:
[tex]g(x)=-f(x-2)+2[/tex]A rule for g is therefore:
[tex]g(x)=-f(x-2)+2[/tex]A triangle has sides with lengths of 20 cm, 15 cm, and 10 cm. Substitute these values into the Pythagorean Theorem to determine if the sides form a right triangle.
Answers
Answer:
10^2 + 15^2 = 20^2
100+225≠400
so no, it's not a right angled triangle
:)
Could someone please help me this is due at 3 pm and it's allready 2:43 and i only need these questions and i am done please help me. if you answer all you get 20 points! PLEASE HELP PLEASE HELP ANYONE PLEASE PLEASE
Answers
a) (2.3 × [tex]$10^4[/tex]) × (1.5 × [tex]$10^{-2}[/tex]) in standard form
The number in standard form is 345
explanation:
The given two multiplication of the numbers is :
(2.3 × [tex]$10^4[/tex]) × (1.5 × [tex]$10^{-2}[/tex])
First we multiply the non exponential terms as shown below
2.3 × 1.5 = 3.45
Now we multiply the exponential terms as shown below
[tex]$10^4[/tex] × [tex]$10^{-2}[/tex] = 100
Now multiplying the exponential and non exponential term we get the number in standard form as shown below
(2.3 × [tex]$10^4[/tex]) × (1.5 × [tex]$10^{-2}[/tex]) in standard form 345
(3.6 × [tex]10^-5[/tex])÷(1.8 × 10²) in standard form 0.0000002
(8 × 10^-3) × (2 × [tex]10^-4[/tex] in standard form 1.6 × [tex]10^-6[/tex]
(6 × 10²)/(3 × 10-5) in standard form 20,00,0000
5.1 × [tex]$10^-1[/tex] in ordinary number is 0.51
(1.7 × [tex]10^4[/tex]) × (8.5x ×[tex]0^-2[/tex]) in standard form 1.445 × 10³.
3.45 x 100 = 345
Therefore, the number in the standard form is 345
b) (3.6 × [tex]10^-5[/tex])÷(1.8 × 10²) in standard form
The number in standard form is 0.0000002
explanation:
By dividing 3.6 by 1.8 we get “2”, hence the above equation becomes,
(2 × [tex]10^-5[/tex]) /[tex]10^2[/tex]
We know that
[tex]\frac{x^{a} }{x^{b} } = x^{a-b}[/tex]
Therefore the above equation becomes,
2 × [tex]10^-7[/tex]
since the exponent of 10 is negative , therefore 7 zeros are written on the left hand side of the number = 0.0000002
Therefore, the number in the standard form is 0.0000002
c) (8 × [tex]10^-3[/tex]) × (2 × [tex]10^-4[/tex]) in standard form
The number in standard form is 1.6 × [tex]10^-6[/tex]
explanation:
Given the product of scientific notations:
(8 × [tex]10^-3[/tex]) × (2 × [tex]10^-4[/tex])
This can be expressed as:
(8 × 2) × ([tex]10^-3[/tex] × [tex]10^-4[/tex])
= 16 × [tex]10^-3[/tex]-4
= 16 × [tex]10^-7[/tex]
= 1.6 × [tex]10^1[/tex] × [tex]10^-7[/tex]
= 1.6 × [tex]10^-6[/tex]
Therefore, the number in the standard form is 1.6 × [tex]10^-6[/tex]
d) (6 × 10²)/(3 × 10-5) in standard form
The number in standard form is 20,00,0000
e) 5.1 × 10^-1 as an ordinary number
The ordinary number is 0.51
usual form of 5.1 × [tex]10^1[/tex] is
5.1 × 1/ [tex]10^1[/tex]
= 5.1/10
= 0.51
Therefore , ordinary number is 0.51
f) (1.7 × [tex]10^4[/tex]) × (8.5 × [tex]10^-2[/tex]) in standard form.
The number in standard form is 1.445 × 10³.
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Aaron has a smart phone data plan that costs $40 per month that
includes 6 GB of data, but will charge an extra $25 per GB over the
included amount. How much would Aaron have to pay in a month
where he used 3 GB over the limit? How much would Aaron have to
pay in a month where he used went over by x GB?
Total cost when over by 3 GB:
Total cost when over by x GB:
Pls give me correct answer
I need it or else I fail
Answers
Aaron has to pay $ 115 when the usage gets over by 3 GBs and
$ (40 + 25x) when the usage gets over by x GBs.
How can one calculate the cost for a multiple of the same entity?
To calculate the cost of a multiple entities when the cost of a single one is given, we linearly multiply a fixed proportion to it to calculate the same.
Given, monthly fixed cost of the smart phone data plan = $ 40
Also, amount charged by the firm for each extra GB used = $ 25
Now, amount charged for three extra GBs = 25*3 = 75
Thus, net total that Aaron has to pay for 9 GBs = 40 + 75 = $ 115
Following similar arguments, amount charged for extra 'x' GBs = 25x
Thus, net total that Aaron has to pay for x GBs = 40 + 25x = $ (40 + 25x)
Therefore, Aaron has to pay $ 115 when the usage gets over by 3 GBs and $ (40 + 25x) when the usage gets over by x GBs.
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1. Alex and chris share sweets
In ratio Alex : Chris= 7:3
Work out the number of
Sweets chris recieves.
Answers
Using ratios, we can conclude that Chris gets 12 sweets.
What are ratios?In mathematics, a ratio shows how many times one number is represented by another. If there are eight oranges and six lemons in a bowl of fruit, the proportion of oranges to lemons would be eight to six. 8:14 for oranges and 6:8 for lemons, respectively, are the ratios of oranges to the total amount of fruit.So, a number of sweets Chris receive:
Alex : Chris = 7:3Let, Chris sweets be 'x' and Alex sweets be 'x + 20'Now, 7:3 = x+20:x, solve as follows:
7:3 = x+20:x7/3 = x+20/x7x = 2x + 607x - 2x = 605x = 60x = 60/5x = 12
Therefore, using ratios, we can conclude that Chris gets 12 sweets.
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The correct question is given below:
Alex and Chris share sweets in the ratio Alex: Chris = 7 : 3. Alex receives 20 more sweets than Chris. Work out the number of sweets Chris receives.
Perform the indicated operation.55.557 + 121.4474 + 6.3
Answers
55.557 + 121.4474 + 6.3
we have that
55.557=55.5570
121.4474
6.3=6.3000
so
55.5570
121. 4474
6. 3000
------------------
183,3044Kuta Software In Systems of Two Solve each system by graphing. 2) y=x+2 1) y=-3x +41 y= 3x - 2
Answers
ok
Equations : 4x + y = 2 x - y = 3
Equation 1 4x + y = 2 x - y = 3
x y x y
-2 10 -2 -5
-1 6 -1 -4
0 2 0 -3
1 -2 1 -2
2 -6 2 -1
The point where both lines cross is (1, -2) so it is the solution.
What’s 1+1 I need to know asap
Answers
Answer:
2
Step-by-step explanation:
convert 1 + the other one is 2
Answer:
2
Step-by-step explanation:
You add both numbers that have been given which are 1+1
The radius of a circle is 18 miles. What is the area of a sector bounded by a 72° arc?Give the exact answer in simplest form. ____ square miles. (pi, fraction,)
Answers
Area = (π*r^2*α)/360°
α =72°
r = 9 mi
Then
A = (π * 81 * 72°)/360° = 81/5π square meters
when a grizzly bear hibernates, its heart rate drops to 101010 beats per minute, which is 20\ , percent of its normal value. what is a grizzly bear's normal heart rate when not hibernating? beats per minute
Answers
A grizzly bear's normal heart rate when not hibernating is 50 beats per minute.
What is a heart rate?
The number of heartbeats that occur in a specific amount of time, usually one minute. The wrist, side of the neck, back of the knees, top of the foot, groin, and other areas of the body where an artery is close to the skin can all detect the heartbeat.
Here,
We let his normal heart rate = h
and given in the question,
20% of h = 10
20/100 * h = 10
h = 10*5
= 50 beats/min
Hence, a grizzly bear's normal heart rate when not hibernating is 50 beats per minute.
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a water tank is to contain 2,000 liters of water it is constructed to be 4 meters long and 1.5 meters wide. find the height of the tank.
Answers
Remember that
1 m3=1,000 lt
2 m3=2,000 lt
The volume of the tank is equal to
V=L*W*H
we have
V=2 m3
L=4 m
W=1.5 m
H=?
substitute given values
2=4*1.5*H
solve for H
2=6H
H=2/6
H=1/3 mI really need help on this!
Answers
Answer:
chad's other friend picked up 1 3/8 pounds of berries
Step-by-step explanation:
chad = 2 1/2 = 2 4/8
friend 1 = 1 3/8
2 4/8 + 1 3/8 = 3 7/8
5 1/4 = 5 2/8
5 2/8 - 3 7/8 = 1 3/8
PLEASE HELP ASAPPP!!!!
The function f(x) is shown on the graph.
The graph shows a downward opening parabola with a vertex at negative 3 comma 16, a point at negative 7 comma 0, a point at 1 comma 0, a point at negative 6 comma 7, and a point at 0 comma 7.
What is the standard form of the equation of f(x)?
f(x) = −x2 − 6x + 7
f(x) = −x2 + 6x + 7
f(x) = x2 − 6x + 7
f(x) = x2 + 6x + 7
Answers
Answer:
f(x)= -x^2-6x+7
Step-by-step explanation:
Since the graph opens downward the function for the parabola is going to be negative. This leaves us with two answers.
Now use the remaining possible equations and solve for the vertex.
Using x= -b/2a to solve for the x value of the vertex plug in the values to solve.
a= -1 b= -6 c=7 so, since -6 is already negative plugging it in makes it positive -(-6)/2(-1) = 6/-2 = -3. So x equals negative 3.
Then plug in -3 into the equation to get the y-value.
- (-3)^2 - 6(-3) + 7 = -9 + 18 + 7 = 16 so this confirms the vertex for this equation is (-3,16) so the answer is -x^2-6x+7.
Hope this helped! :)
1/3 divided by 2 (1/2)^3
please help me
Answers
Answer:
1/24
Step-by-step explanation:
.........................
Answer:
I'll assume that "2 (1/2)^3" means 2*(1/2)^3.
Step-by-step explanation:
(1/3)/(2*(1/2)^3)
(1/3)/(1)^3
= (1/3)
===========================
If "2(1/2)^3" means (2+1/2)^2:
((5/2)^2 = (25/4)
--
(1/3)/(25/4)
(1/3)*(4/25) = 4/75 or 0.05333
If the interest earned on an account after 2 years is 15$ how much would it be after 10 years? why?
Answers
The amount of interest earned in 10 years is $75.
What is interest?Interest is the sum over and above the original amount (the amount borrowed) that is paid to a lender or depositor by a borrower or deposit-taking financial institution at a set rate. It is not the same as a fee that the borrower might pay to the lender or another entity.It also differs from a dividend, which is money given to shareholders (owners) by a company from its profit or reserve but not at a set rate predetermined in advance, but rather on a pro-rata basis as a share of the reward received by risk-taking business people when revenue is earned that exceeds all costs.So, interest earned after 2 years = $15
interest earned in 1 year = 15/2 = $7.5 interest earned in 10 years = 10 × $7.5= $ 75Therefore, the amount of interest earned in 10 years is $75.
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-11x-y+3z=-21
(5x-y-5z=-5)
(3x+y+z=13)
Answers
Answer: [tex]y=15-5x, z=2x-2[/tex]
Step-by-step explanation:
Subtracting the first equation from the second equation yields [tex]16x-8z=16 \implies 2x-z=2 \implies z=2x-2[/tex].
Substituting this into the first equation,
[tex]-11x-y+3(2x-2)=-21\\\\-11x-y+6x-6=-21\\\\-5x-y-6=-21\\\\-5x-y=-15\\\\5x+y=15\\ \\ y=15-5x[/tex]
If f(x) is an exponential functionwhere f(-2) = 1 and f(7) = = 63,then find the value of f(1) , to thenearest hundredth.
Answers
An exponential function has the form
[tex]y=ab^x[/tex]Therefore, to find an exponential function that satisfies our condition, we need to find a and b.
From f(-2) = 1, we have
[tex]1=ab^{-2}\: ^{}\: \: \: \: \; ^{}\: \: \: \: \; (1)[/tex]and from f(7) = 63, we have
[tex]63=ab^7\: \: \: \: \; ^{}\: \: \: \: \; (2)[/tex]Solving for a in equation (1) gives
[tex]a=b^2[/tex]substituting this value of a into equation (2) gives
[tex]63=b^2\cdot b^7[/tex][tex]63=b^8[/tex][tex]\begin{gathered} \therefore b=\sqrt[8]{63} \\ b=1.6785 \end{gathered}[/tex]With the value of b in hand, we now find the value of a:
[tex]\begin{gathered} a=b^2 \\ \therefore a=2.8173 \end{gathered}[/tex]Hence, the exponential function is
[tex]f(x)=(2.8173)(1.6785)^x[/tex]Evaluating the above function at x = 1 gives
[tex]\begin{gathered} f(1)=(2.8173)(1.6785)^1 \\ \boxed{\therefore f(1)=4.73.} \end{gathered}[/tex]which is our answer!
a company produces very unusual CD's for which the variable cost is 8$ per CD and the fixed costs are $30000. They will sell the CD's for $63 each. Let x be the number of CD's produced. Write the total cost C as a function of the number of CD's produced.C=$___Write the total revenue R as a function of the number of CD's produced R=$_______Write the total profit P as a function of the number of CD's producedP=$______Find the number of CD's which must be produced to break even.The number of CD's which must be produced to break even is_______
Answers
Remember that to model this kind of problems, we can use the equation of a straight line
[tex]y=mx+b[/tex]Where:
• b, are the fixed costs
,• x ,represents what varies
,• m, is the cost of what varies
1. Therefore, using the data provided, we can model the cost of production with a function where:
• x ,is the numbers of CD's produced
,• C, is the cost of producing those CD's
[tex]C=8x+30000[/tex]2. To model the revenue, we'll have to take into account that every CD sells at $68. Therefore, the money collected by selling the x CD's produced is:
[tex]R=63x[/tex]Where:
• x ,its the number of CD's produced and sold
,• R, is the total revenue
3. To get the profit, we'll have to substract the production cost to the money collected by the sale. Remember we already have two expressions, in terms of x, that describe both situations. Therefore,
[tex]\begin{gathered} P=R-C \\ \rightarrow P=63x-(8x+30000)\rightarrow P=63x-8x-30000 \\ \rightarrow P=55x-30000 \end{gathered}[/tex]Where:
• x ,its the number of CD's produced and sold
,• P, is the total revenue
4. In order to break even, the revenue (R) has to be equal to the cost of production (C), Thus,
[tex]\begin{gathered} R=C\rightarrow63x=8x+30000 \\ \rightarrow63x-8x=30000 \\ \rightarrow55x=30000\rightarrow x=\frac{30000}{55} \\ x=545.5 \\ \text{Rounding to closest integer,} \\ x=546 \end{gathered}[/tex]Therefore, 546 CD's would have to be produced to break even.
I would really appreciate your help
Answers
Answer:
The answer is 35
Step-by-step explanation:
Pls mark this answer as brainliest
Given v = 60sinθ, what is the instantaneous voltage when θ = 30⁰?Question 11 options:6034.643051.96
Answers
30
Explanations:
Given the expression for the instantaneous voltage expressed as:
[tex]v=60\sin \theta[/tex]Given the following parameter:
θ = 30⁰
Substitute the given parameter into the formula to have:
[tex]\begin{gathered} v=60\sin 30^0 \\ v=60(0.5) \\ v=30 \end{gathered}[/tex]Therefore the instantaneous voltage when θ = 30⁰ is 30.