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
Lerok [7]
3 years ago
14

Find all the zeros of the equation x^4-6x^2-7x-6=0 Explain please.

Mathematics
1 answer:
Alina [70]3 years ago
8 0
<h3>Answer:</h3>
  • zeros are {-2, 3, (-1±i√3)/2}
<h3>Step-by-step explanation:</h3>

I like to look at a graph of the function to see where the zeros might be. Here, there are x-intercepts at x=-2 and x=3. These can be factored out using synthetic division to find the factorization to be ...

... (x +2)(x -3)(x² +x +1) = 0

By completing the square, using the quadratic formula, or by looking at the graph of it, the complex roots of the quadratic factor can be found to be ...

... x = (-1 ±i√3)/2

_____

The second attachment shows my synthetic division. The first division takes out the root x=3 to give a quotient of x³ +3x² +3x +2. The second division takes out the root -2 to give the quotient of x² +x +1. (You can see that I tried -1 as a root first.)

The graph shows both the quartic and the quadratic factor of it. The latter has a leading coefficient of 1 and a vertex at (-1/2, 3/4), so you know the complex roots are -1/2 ±i√(3/4).

_____

<em>From the beginning</em>

There is only a very complicated formula for the roots of a quartic equation, so these are usually solved by machine or by some form of trial and error (iteration). There are some helps, like Descarte's Rule of Signs, and the Rational Root theorem.

Here, the former looks at the one sign change in the coefficients to tell you there will be 1 positive real root. Changing the sign of the odd-degree terms makes there be 3 sign changes, so there will be 3 or 1 negative real roots. Thus, we're assured at least two real roots, one of each sign.

We can look at the constant term to find the y-intercept to be -6. We can add the coefficients to find the value of the function is -18 for x=1, so the positive real root is larger than 1.

The Rational Root theorem says any rational roots will be factors of 6, the constant term. Choices are 1, 2, 3, 6. We have already eliminated 1 as a possibility, and we consider it unlikely that 6 will be a root. (The 4th power overwhelms the other terms in the function.) We tried 2 and found it doesn't work (this was before we graphed the function). The attached division result shows that 3 is a root, as does the graph.

Once you get down to a quadratic, you can find the remaining roots in the usual way. Because it is so simple to read them from the graph, we decided to graph the quadratic factor.

_____

<em>Comment on terminology</em>

"root" and "zero" are essentially the same thing when the function is equated to zero, as here. The terms refer to the value(s) of x that make the polynomial function evaluate to zero.

You might be interested in
Solve for x. Show each step of the solution. Show all work. 4.5(8-x)+36-2.5(3x+24)=102
sergejj [24]

4.5(8 - x) + 36 = 102 - 2.5(3x + 24)

36 - 4.5x + 36 = 102 - 7.5x - 60

72 - 4.5x = 42 - 7.5x

-4.5x + 7.5x = 42 - 72

3x = - 30

x = -30/3

x = -10 <===

Hope this helps!

4 0
3 years ago
What is the probability of choosing a king from a deck of cards? Type your answer as a fraction.
Vikki [24]

Answer:

4/52 or 1/13

Step-by-step explanation:

there are only 4 kings; diamond, Heart, Spades and Clubs.

4 0
3 years ago
Read 2 more answers
Share $600 between Anna and Raman in the ratio 3:7 and 1:4
Ulleksa [173]

Answer:

3 : 7 =>

Anna = $180

Raman = $420

1 : 4 =>

Anna = $120

Raman = $480

Step-by-step explanation:

<u>Shares when ratio is 3 : 7</u>

Anna : Raman = 3 : 7

Sum of ratio 10

Anna's share=

                   \frac{3}{10} \times 600 = \$  180

Raman 's share=

                  \frac{7}{10} \times 600 = \$ 420

<u>Shares when ratio is 1 : 4</u>

Anna : Raman = 1 : 4

Sum of ratio = 5

Anna 's share =

                   \frac{1}{5} \times 600 = \$ 120

Raman's share =

                  \frac{4}{5} \times 600 = \$ 480

8 0
3 years ago
PLSSSS HELPPP THANK YOOOUU!!Let f(x) = 3x + 5 and g(x) = 2x - 1 Perform the following function operation (if needed use ^ to mea
jek_recluse [69]

Answer:

I hope it will help you....

5 0
3 years ago
HELP ME I MARK BRAIN THING
disa [49]

Answer:

your answer will be B

Step-by-step explanation:

have a nice day/night !!!

7 0
2 years ago
Read 2 more answers
Other questions:
  • Determine whether each of the following functions is a solution of Laplace's equation uxx + uyy = 0. (Select all that apply.) u
    5·1 answer
  • Classify the scatter plot as having a negative or no correlation. HOMEWORK DUE TOMORROW
    12·1 answer
  • A plant is already 14 centimeters tall, and it will grow one centimeter every month.
    11·1 answer
  • Sarafina and Katie have between $105 and $420 dollars to spend on jewelry for Christmas presents for their friends. If they buy
    8·1 answer
  • Order from least to greatest 14%, 1/5, 0.03
    9·2 answers
  • The formula Upper A equals 23.1 e Superscript 0.0152 tA=23.1e0.0152t models the population of a US​ state, A, in​ millions, t ye
    12·1 answer
  • How many skimboards and longboards must the shop sell each week to make a profit
    14·2 answers
  • Pls answer i’m begging
    13·2 answers
  • CD is perpendicular to AB CREATE AC AND BC AND MEASURE THEIR LENGTHS
    6·1 answer
  • What is the standard form of 0.00849​
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!