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 algebraic expression for the phrase "thirty less than the product of a number y and four".
zzz [600]

Answer:

30-(y*4)

Step-by-step explanation:

replace thirty with 30. less than is - (minus). the product of a number y and four. replace four with 4. the product is the answer of the multiplication problem. so the multiplication problem here would be y*4. so the full answer is

30-(y*4)

8 0
3 years ago
Read 2 more answers
What is 2×2 +34 = what is one hundred and twenty seven in numbers<br>​
jarptica [38.1K]
38 and 127
Hope this helps
5 0
3 years ago
Read 2 more answers
Plz answer this!!! A triangle with sides 11 m , 13 m and 18 m is a right triangle. True or false
Nata [24]

Answer:

It depends on what sides equal which measurement. But it could be true.

6 0
3 years ago
A. 2x4 + 21x2 +3x -2<br>B. 2x4 + 9x2 -2<br>C. 3x2 + 6x+ 1 <br>D. 2x4 + 9x3 +12x2 + 3x -2 <br> ​
Inga [223]
You need to foil, The answer is D
6 0
3 years ago
How do you write your answer as a mixed number fraction?
alexandr402 [8]

Answer:

to convert an improper fraction to a mixed fraction, follow these steps:

-divide the numerator by the denominator.

-write down the whole number answer.

-then write down any remainder above the denominator.

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Other questions:
  • What is the answer to 1/7(x+3/4)=1/8
    6·1 answer
  • Your cellphone plan gives you 500 minutes monthly. If you have used 310 minutes already and there is a week left this month, how
    12·2 answers
  • Help me what is the answer ?
    5·2 answers
  • How many solutions does
    5·1 answer
  • What is the product? 4 • (–5)
    9·2 answers
  • A flight from houston to chicago is approximately 940 miles. A plane making this flight is traveling at an average rate of 485 m
    10·1 answer
  • Consider the following data on 20 chemical reactions, with Y = chromatographic retention time (seconds) and X = molecular weight
    8·1 answer
  • What is the equivalent ratio for 3:4​
    14·2 answers
  • ILL BRAINLIEST YOU IF YOU GET IT RIGHT<br><br> explanation please
    10·2 answers
  • Help? I need a answer
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!