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4 years ago

How many roots of the polynomial 30x^4 + 7x^3-125x^2-54x+72 have absolute value greater than 1?

Mathematics

1 answer:

Daniel [21]4 years ago

8 0

Answer:

There are 3 roots of the given polynomial that have absolute value greater than 1.

Step-by-step explanation:

We are given the polynomial:

$p(x) = 30x^4 + 7x^3-125x^2-54x+72$

We can factorize the given polynomial as:

$p(x) = 30x^4 + 7x^3-125x^2-54x+72\\=(x-2)(2x+3)(5x-3)(3x+4)=0$

$\text{Solving for}\\(x-2) = 0\\x = 2\\|x| = 2 > 1$

$\text{Solving for}\\(2x+3) = 0\\x = \dfrac{-3}{2}\\\bigg|\dfrac{-3}{2}\bigg| = \dfrac{3}{2} > 1$

$\text{Solving for}\\(5x-3) = 0\\x = \dfrac{3}{5}\\\bigg|\dfrac{3}{5}\bigg| = \dfrac{3}{5} < 1$

$\text{Solving for}\\(3x+4) = 0\\x = \dfrac{-4}{3}\\\bigg|\dfrac{-4}{3}\bigg| = \dfrac{4}{3} > 1$

Thus, there are 3 roots of the given polynomial that have absolute value greater than 1.

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saveliy_v [14]

Given:

The two sides of a triangle are 2 and 6.

To find:

The side length that would NOT work to make a triangle with the two slide lengths of 2 and 6.

Solution:

For a triangle, the sum of two smaller sides must be greater than the largest side.

Let x be the third side.

Case 1: x is the largest side, then

$2+6>x$

$8>x$ ...(i)

Case 2: x is not the largest side, then

$2+x>6$

$x>6-2$

$x>4$ ...(ii)

From (i) and (ii), we get

$4$

Here, 4 and 8 are not included.

6, 5, 7 are included in the above interval, but 4 is not included in the above interval. It means 4 would NOT work to make a triangle with the two slide lengths of 2 and 6.

Therefore, the correct option is D.

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3 years ago

What does 5/9-1/12 equal to?

Elena-2011 [213]

$\frac{5}{9} - \frac{1}{12} = \frac{5 \times 4}{9 \times 4} - \frac{1 \times 3}{12 \times 3}$
$= \frac{ 20}{36} - \frac{3}{36}$
$= \frac{17}{36}$

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4 years ago

In which quadrant is the point (4,-2 located<br>

zheka24 [161]

Answer:

quadrant 4

Step-by-step explanation:

Hope this helped :)

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4 0

3 years ago

Read 2 more answers

X + 2 > 5.8 Whats The VALUE OF X

Oduvanchick [21]

Answer:

X > 3.8

Step-by-step explanation:

x +2 > 5.8

x +2 -2 > 5.8 -2

x > 3.8

7 0

3 years ago

Read 2 more answers

A recent study focused on the number of times men and women send a WeChat message in a day. The information is summarized next.

Ratling [72]

Answer:

1)Null hypothesis: $\mu_{men}=\mu_{women}$

Alternative hypothesis: $\mu_{men} \neq \mu_{women}$

2) Two critical values are $z_{\alpha/2}=-2.58$ and $z_{1-\alpha/2}=2.58$

3) $z=-2.40$

4) $p_v =2*P(z$

5) Comparing the p value with the significance level given $\alpha=0.01$ we see that $p_v>\alpha$ so we can conclude that we FAIL to reject the null hypothesis, and a would NOT be a significant difference in the mean number of times men and women send a Twitter message in a day.

Step-by-step explanation:

Data given and notation

$\bar X_{men}=23$ represent the mean for the sample men

$\bar X_{women}=28$ represent the mean for the sample women

$\sigma_{men}=5$ represent the population standard deviation for the sample men

$\sigma_{women}=10$ represent the population standard deviation for the sample women

$n_{men}=25$ sample size for the group men

$n_{women}=30$ sample size for the group women

t would represent the statistic (variable of interest)

$\alpha=0.01$ significance level provided

1）Develop the null and alternative hypotheses for this study?

We need to conduct a hypothesis in order to check if the means for the two groups are different, the system of hypothesis would be:

Null hypothesis: $\mu_{men}=\mu_{women}$

Alternative hypothesis: $\mu_{men} \neq \mu_{women}$

Since we know the population deviations for each group, for this case is better apply a z test to compare means, and the statistic is given by:

$z=\frac{\bar X_{men}-\bar X_{women}}{\sqrt{\frac{\sigma^2_{men}}{n_{men}}+\frac{\sigma^2_{women}}{n_{women}}}}$ (1)

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

2）Determine the critical value(s).

Based on the significance level $\alpha=0.01$ and $\alpha/2=0.005$ we can find the critical values with the normal standard distribution, we are looking for values that accumulates 0.005 of the area on each tail on the normal distribution.

For this case the two values are $z_{\alpha/2}=-2.58$ and $z_{1-\alpha/2}=2.58$

3) Calculate the value of the test statistic for this hypothesis testing.

Since we have all the values we can replace in formula (1) like this:

$z=\frac{23-28}{\sqrt{\frac{5^2}{25}+\frac{10^2}{30}}}}=-2.40$

4）What is the p-value for this hypothesis test?

Since is a bilateral test the p value would be:

$p_v =2*P(z$

5）Based on the p-value, what is your conclusion?

Comparing the p value with the significance level given $\alpha=0.01$ we see that $p_v>\alpha$ so we can conclude that we FAIL to reject the null hypothesis, and a would NOT be a significant difference in the mean number of times men and women send a Twitter message in a day.

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3 years ago