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julsineya [31]
3 years ago
6

Apply The Remainder Theorem, Fundamental Theorem, Rational Root Theorem, Descartes Rule, and Factor Theorem to find the remainde

r, all rational roots, all possible roots, and actual roots of the given function. Make Sure to graph the function and state the turning points (Relative maximum and relative minimum) and end behavior (Graph is rising or falling on the right or left side, etc.)
f(x) = 3x^3 − x^2 + 8x + 12
Please answer all the questions fully asap!!
Mathematics
1 answer:
Over [174]3 years ago
4 0

9514 1404 393

Answer:

  possible rational roots: ±{1/3, 2/3, 1, 4/3, 2, 3, 4, 6, 12}

  actual roots: -1, (2 ±4i√2)/3

  no turning points; no local extrema

  end behavior is same-sign as x-value end-behavior

Step-by-step explanation:

The Fundamental Theorem tells us this 3rd-degree polynomial will have 3 roots.

The Rational Root Theorem tells us any rational roots will be of the form ...

  ±{factor of 12}/{factor of 3} = ±{1, 2, 3, 4, 6, 12}/{1, 3}

  = ±{1/3, 2/3, 1, 4/3, 2, 3, 4, 6, 12} . . . possible rational roots

Descartes' Rule of Signs tells us the two sign changes mean there will be 0 or 2 positive real roots. Changing signs on the odd-degree terms makes the sign-change count go to 1, so we know there is one negative real root.

The y-intercept is 12. The sum of all coefficients is 22, so f(1) > f(0) and there are no positive real roots in the interval [0, 1]. Synthetic division by x-1 shows the remainder is 22 (which we knew) and all the quotient coefficients are all positive. This means x=0 is an upper bound on the real roots.

The sum of odd-degree coefficients is 3+8=11, equal to the sum of even-degree coefficients, -1+12=11. This means that -1 is a real root. Synthetic division by x+1 shows the remainder is zero (which we knew) and the quotient coefficients alternate signs. This means x=-1 is a lower bound on real roots. The quotient of 3x^2 -4x +12 is a quadratic factor of f(x):

  f(x) = (x +1)(3x^2 -4x +12)

The complex roots of the quadratic can be found using the quadratic formula:

  x = (-(-4) ±√((-4)^2 -4(3)(12)))/(2(3)) = (4 ± √-128)/6

  x = (2 ± 4i√2)/3 . . . . complex roots

__

The graph in the third attachment (red) shows there are no turning points, hence no relative extrema. The end behavior, as for any odd-degree polynomial with a positive leading coefficient, is down to the left and up to the right.

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pashok25 [27]

Answer:

  1. So you know that the height that is visible above x=0 is 74.98 m
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That tells us that we have two parts account for

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8 0
2 years ago
Read 2 more answers
The table below shows the numbers of tickets sold at a movie theater on Friday.
sleet_krkn [62]

Answer:

Number of Adult's tickets sold on Saturday = 3,356

Number of Children's tickets sold on Saturday = 2, 928

Total number of tickets sold over these two days is 8,938.

Step-by-step explanation:

Here, the number of tickets sold on FRIDAY:

Adult Ticket sold = 1,678

Children's Tickets sold = 976

So, the total number of tickets sold on Friday  

= Sum of ( Adult + Children's ) tickets  = 1,678  + 976 = 2,654 ....  (1)

The number of tickets sold on SATURDAY:

Adult Ticket sold =   2 times as many as the number of adult tickets sold

on Friday

=  1,678 x 2  = 3,356

Children's Tickets sold = 3 times as many as the number of children's

tickets sold on Friday.

=  976  x 3 = 2, 928

So, the total number of tickets sold on Saturday  

= Sum of ( Adult + Children's ) tickets  = 3,356 + 2,928 = 6, 284 ....  (2)

Now, the total number of tickets booked in these two days :

Sum of tickets booked on (Friday + Saturday)

= 2,654 +  6, 284  =   8,938

Hence, total number of tickets sold over these two days is 8,938.

7 0
3 years ago
The height of a triangle is 4 times its base. if he are of a triangle is 200cm, what is the height
uranmaximum [27]
A = (h * b) / 2
A = 200
h = 4b

200 = (b(4b) / 2
200 * 2 = b * 4b
400 = 4b^2
400 / 4 = b^2
100 = b^2
sqrt 100 = b
10 = b ..... base is 10 cm

h = 4b
h = 4(10)
h = 40 <=== height is 40 cm
3 0
3 years ago
A recipe calls for 5 8 pound of cheese per batch. Assuming you have enough of the other ingredients, how many batches of the rec
aliya0001 [1]
I assume you mean 5/8 of a pound not 58 pounds. In that case convert your total number of pounds into eights to make it easier to calculate. 5 pounds is equal to 40/8 pounds of cheese. Now divide this by 5/8. That equals 8. SO you can make 8 batches.
3 0
3 years ago
How to answer this question
Sonbull [250]
Look at picture for details

6 0
3 years ago
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