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
Dovator [93]
3 years ago
8

How do you solve x^4 - 3x^3 - 3x^2 - 75x - 700

Mathematics
1 answer:
madam [21]3 years ago
4 0

Answer:

  = (x +4)(x -7)(x^2 +25)

  roots are -4, 7, ±5i

Step-by-step explanation:

You have not said what "solve" means in this context. An expression by itself doesn't have a solution. We have assumed you want to find the factoring and/or roots of it.

I like to use a graphing calculator to find the real roots. For this expression, there are two of them, one positive and one negative. (You know there will be one positive real root, and at least one negative real root, from Descartes's rule of signs.)

Then those roots can be factored out and the solution to the remaining quadratic determined. That factoring can occur by polynomial long division, synthetic division, or other means.

I like to see what happens when I plot the graph of the function divided by the known factors. (We expect a parabola that doesn't cross the x-axis.) The vertex of that parabola can be used to find the remaining roots.

The x-intercepts of the given expression are -4 and +7, so two of the factors are (x+4) and (x-7). Dividing these from the given expression (by synthetic division or other means) gives (x^2 +25). This only has imaginary roots (±5i).

____

If you're constrained to doing this "by hand" with only a scientific calculator, Descartes's rule of signs tells you there is one positive real root. (Only one sign change in the sequence of coefficient signs: +----.)

The rational root theorem tells you it will be a divisor of 700. Various estimates of the maximum magnitude of it will tell you it is probably less than 14 (easily checked). So, the numbers you can test as roots would be 1, 2, 4, 5, 7, 10, 14. You will find that 7 is a root, and then you can reduce the problem to the cubic x^3 +4x^2 +25x +100.

When odd-degree term signs are changed, there will be 3 sign changes (-+-+), hence at least one negative real root. The rational root theorem tells you it is a divisor of 100, so possible choices are -1, -2, -4, -5. By trial and error or other means, you can find the root to be -4. Then the problem reduces to the quadratic x^2 +25.

Roots of that are ±√(-25) = ±5i.

This process generally entails a fair amount of trial-and-error work, which is why I prefer one that makes some use of technology.

_____

We have presumed you have some familiarity with ...

  • Descartes's rule of signs
  • Rational Root Theorem
  • synthetic division

This will usually be the case when you're presented with problems like this. If you need additional information on any of these, it is readily available on the internet (and probably also in your reference material).

You might be interested in
Write an equation of the parabola that opens up whose vertex (−1, 2) is 3 units from the focus. (Vertex form or Parabola Form is
TEA [102]

Answer:

y2  =  4ax (opens right, a > 0)

y2  =  -4ax (opens right, a > 0)

x2  =  4ay (opens up, a > 0)

x2  =  -4ay (opens down, a > 0)

Vertex at (h, k) :  

(y - k)2  =  4a(x - h) (opens right, a > 0)

(y - k)2  =  -4a(x - h) (opens right, a > 0)

(x - h)2  =  4a(y - k) (opens up, a > 0)

(x - h)2  =  -4a(y - k) (opens down, a > 0)

Equation of a Parabola in Vertex form

Vertex at Origin :  

y  =  ax2 (opens up, a > 0)

y  =  -ax2 (opens down, a > 0)

x  =  ay2 (opens right, a > 0)

x  =  -ay2 (opens left, a > 0)

Vertex at (h, k) :  

y  =  a(x - h)2 + k (opens up, a > 0)

y  =  -a(x - h)2 + k (opens down, a > 0)

x  =  a(y - k)2 + h (opens right, a > 0)

y  =  -a(y - k)2 + h (opens left, a > 0)

Step-by-step explanation:

7 0
2 years ago
A Chain of Events
OverLord2011 [107]
Each person should ride the bike for 4 miles, so that every one has to walk 8 miles and ride 4 miles.

If they had 2 bikes, each person will ride for 8 miles, 4 miles on each bike, and then walk 4 miles.

5 0
3 years ago
BIBIDI BABIDI BUU I NEED HALP FROM YUU
Leokris [45]

Answer:

The answer is B. IV

7 0
2 years ago
What is the range of the following exponential function?plz help asap
IrinaVladis [17]

Answer:

0.23

Step-by-step explanation:

5 0
2 years ago
Jason can travel 24 3/4 miles in one half hour what is his average speed in miles per hour
BARSIC [14]
Average speed = 24 3/4 / 1 1/2 = 99/4 / 3/2 = 99/4 x 2/3 = 16 1/2 miles per hour.
3 0
3 years ago
Read 2 more answers
Other questions:
  • There are five shelves of video games in a video store. There are six video games on each shel. how many video games are there o
    5·2 answers
  • Find the constant of variation k for the direct variation
    6·1 answer
  • Roger read 4 more books in July then he did in January how many books did he read in July
    5·1 answer
  • Which expression is equivalent to 4(3−6x)+x ?
    15·2 answers
  • What is the length of the line?
    14·2 answers
  • Explain which is faster: 10 centimeters/5 seconds or 10 centimeters/20 seconds.
    8·2 answers
  • Look at the picture and plzz help me
    15·2 answers
  • James and Eric each mix 6 pitchers of lemonade. James makes 26 fluid Ounces of lemonade per pitcher, and Eric makes 32 fluid Oun
    10·1 answer
  • 7. 70% of what number is 112
    7·1 answer
  • Circle A has an area of 500 in? The diameter of circle B is three times the diameter of circle A. Estimate the area of circle B.
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!