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Lana71 [14]
3 years ago
14

Let f be a functions of degree 4 whose coefficients are real numbers: two of its zeros are - 3 and 4 - i. Explain why one of the

remaining zeros must be a real number. Write down one of the missing zeros.
Mathematics
1 answer:
kvasek [131]3 years ago
8 0

Answer:

Step-by-step explanation:

We have the following theorem, if f is a polynomial with real coefficients, we can factor it completely in factors of the degree at most 2.

Consider first a polynomial of degree two, hence it is a polynomial of the form ax^2+bx+c. The cuadratic formula tells us that the solutions are of the form

x = \frac{-b\pm \sqrt[]{b^2-4ac}}{2a}.

Note that square root, over the reals, tells us that the are only real solutions if b^2-4ac \geq 0. If that is not the case, say it's negative, the solution are complex. Then, the solutions are of the form

x = \frac{-b \pm i \sqrt[]{4ac-b^2}}{2a}. NOte that this means that if we have a complex number of the form a+bi that is a solution, then the number a-bi (who is called the complex conjugate) is also a solution.

Recall that when we have a polynomial f(x) whose a zero is the number c, then we can factor f as follows f(x) = (x-c) * p(x) where p(x) is another polynomial of lesser degree .

So far, we know that -3 and 4-i are zeros of the function f. Note that we are missing two zeros. But, since complex numbers are zeros of polynomial only by pairs (that is the number and its conjugate are zeros), then, we must have that one of the missing 2 zeros is a real number. We have 4-i as a zero, then, its complex conjugate must be also a zero, i.e 4+i is a zero.

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The point (8,2,9) is in the first octant.
The closest distance from the coordinate planes is the value y=2, which will therefore be the maximum radius of the required sphere.

The standard equation of a sphere with radius r and centre (u,v,w) is given by
(x-u)^2+(y-v)^2+(z-w)^2=r^2

Substituting r=2, u=8, v=2, w=9, we get the equation of the required sphere
(x-8)^2+(y-2)^2+(w-9)^2=2^2
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An advertisement for a popular weight-loss clinic suggests that participants in its new diet program lose, on average, more than
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Testing the hypothesis, it is found that:

a)

The null hypothesis is: H_0: \mu \leq 10

The alternative hypothesis is: H_1: \mu > 10

b)

The critical value is: t_c = 1.74

The decision rule is:

  • If t < 1.74, we <u>do not reject</u> the null hypothesis.
  • If t > 1.74, we <u>reject</u> the null hypothesis.

c)

Since t = 1.41 < 1.74, we <u>do not reject the null hypothesis</u>, that is, it cannot be concluded that the mean weight loss is of more than 10 pounds.

Item a:

At the null hypothesis, it is tested if the mean loss is of <u>at most 10 pounds</u>, that is:

H_0: \mu \leq 10

At the alternative hypothesis, it is tested if the mean loss is of <u>more than 10 pounds</u>, that is:

H_1: \mu > 10

Item b:

We are having a right-tailed test, as we are testing if the mean is more than a value, with a <u>significance level of 0.05</u> and 18 - 1 = <u>17 df.</u>

Hence, using a calculator for the t-distribution, the critical value is: t_c = 1.74.

Hence, the decision rule is:

  • If t < 1.74, we <u>do not reject</u> the null hypothesis.
  • If t > 1.74, we <u>reject</u> the null hypothesis.

Item c:

We have the <u>standard deviation for the sample</u>, hence the t-distribution is used. The test statistic is given by:

t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}

The parameters are:

  • \overline{x} is the sample mean.
  • \mu is the value tested at the null hypothesis.
  • s is the standard deviation of the sample.
  • n is the sample size.

For this problem, we have that:

\overline{x} = 10.8, \mu = 10, s = 2.4, n = 18

Thus, the value of the test statistic is:

t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}

t = \frac{10.8 - 10}{\frac{2.4}{\sqrt{18}}}

t = 1.41

Since t = 1.41 < 1.74, we <u>do not reject the null hypothesis</u>, that is, it cannot be concluded that the mean weight loss is of more than 10 pounds.

A similar problem is given at brainly.com/question/25147864

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