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WINSTONCH [101]
3 years ago
12

1846 What is the smallest and largest?

Mathematics
1 answer:
aksik [14]3 years ago
7 0
8 is largest and 1 is smallest number
Step by
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Find all the zeros of the equation x^4-6x^2-7x-6=0 Explain please.
Alina [70]
<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.

8 0
3 years ago
Plsss help. Thanks!!
slava [35]

Answer:

401

Step-by-step explanation:

1. Approach

To solve this problem, one first has to think about the given figure in a certain way. In the figure, one can see that it is a circle attached on either side of a rectangle. To find the perimeter of the figure, one has to find the circumference of the circle and then add two sides of the rectangle to the answer

2. Circumference of the circle

The formula to find the circumference of a circle is;

2r(pi) or d(pi)

~ diameter times the value (pi)

Normally to find the circumference of a semicircle, one would have to divide this formula by 2, but since in this case, one has to add two congruent semicircles, so therefore, the effect of dividing the equation by two, only to multiply by two again cancels, and hence, there is no need to divide by 2.

Substitute in the values;

(78)(pi)

~ 245

3. Find the perimeter of the entire object

Now, one has to add the two additional sides of the figure, to the circumferences of the semicircles to get the final answer;

78 + 78 + 245

= 401

3 0
2 years ago
Read 2 more answers
3+1+2+4+2+2+1+4+2+2+2+11+2+2+1+2+1111111111
sertanlavr [38]

Answer:

1,111,111,154

Explanation:

<em>Willing to help, hope you're having a splendiferous day...</em>

<em>Willing to help, hope you're having a splendiferous day... </em>

5 0
2 years ago
Read 2 more answers
The contrapositive of p q is:
Kay [80]
Some symbols are missing. I'm assuming you meant to put in a horseshoe symbol like so p \supset q

If so, then the contrapositive of that would be \sim q \supset \ \sim p which is the final answer. 

The tilde sign \sim means "not" so writing \sim q means "not q" which is the complete opposite of q. 
6 0
3 years ago
A baby gains 11 pounds in its first year of life. The baby gained 45 pounds during the first four months and 3 pounds in
kifflom [539]

Answer:

4 pound is the right answer

Mark me as a brilliant

6 0
2 years ago
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