1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
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.

You might be interested in
Round 7.38 to the nearest tenth
Anna71 [15]

Answer:

7.40

Step-by-step explanation:

7.38, we go to the tenths place and the number after then tenths place five or more u raise the score since its more u change hte score of the 3 to 4 and put a zero in the hundredths place

4 0
3 years ago
Read 2 more answers
Can someone please help me? Please show your work for each answer. Thank you in advance :)
sergey [27]

Answer:

6. c = 5

7. b = 8

8. a = 0.75

9. b = 12

10. c = 0.5

11. c = 17

12. b = 1.34

13. a = 4.5

14. a = 1.2

15. c = 6.1

16. b = 15

17. c = 41

18. a = 7

19. c = 8.5

20. b = 4

Step-by-step explanation:

You need to use Pythagorean theorem. Put the number for a in a, b in b, and c in c. DON'T forget to square a, b, and c.

7 0
3 years ago
Read 2 more answers
2. A triangular pyramid begins with these coordinates:
vagabundo [1.1K]
The rule for the reflection is (x,y,z) → (-x,y,z) so P' = (1, 2, 0) for example. 
<span>Then the rule is (x,y,z) → (x + 2, y + 4, z - 3) so P'' = (3, 6, -3) </span>
3 0
3 years ago
Melanie finds paint at the store for $4.95 per gallon. She also finds a Paintbrush for
tiny-mole [99]

Answer:

y = 4.95x + 3.75

Step-by-step explanation:

3 0
3 years ago
Round 2063 658.497 to the nearest hundred
Soloha48 [4]

Answer:2063 658.5

Step-by-step explanation:

3 0
3 years ago
Other questions:
  • On
    5·2 answers
  • What property is =x+2-2
    8·1 answer
  • An 16 16​-oz jar of peanut butter in the shape of a right circular cylinder is 7 7 in. high and 5 5 in. in diameter and sells fo
    11·1 answer
  • In the number 8444 what is the relationship betwee n the 4s
    8·1 answer
  • The first few steps in deriving the quadratic formula are shown. A table listing the first few steps in deriving the quadratic f
    16·1 answer
  • What is the value of log4_16?​
    15·1 answer
  • Anyone knowwwwwww? by chance
    13·1 answer
  • The length of a city block running north to south in New York City is about 5x10^-2 miles. The distance from New York City to Mu
    13·1 answer
  • 12x-13x write an equivalent expression.
    6·1 answer
  • Please help I am very confused!
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!