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Mashutka [201]
2 years ago
12

How do I use the Descartes Rule of Signs to determine all possible number of positive and negative zeros?

Mathematics
1 answer:
larisa [96]2 years ago
8 0

Answer:

We conclude that:

  • 2  or 0 positive real roots
  • 2  or 0 negative real roots

Step-by-step explanation:

Descartes Rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is:

  • Either equal to the number of sign differences between consecutive nonzero coefficients,
  • Or is less than it by an even number.

Given the function

f\left(x\right)=x^4-2x^3-4x^2+2x+3

So, the coefficients are 1, −2, −4, 2, 3

As can be seen, there are 2 changes.

This means that there are 2 or 0 positive real roots.

To find the number of negative real roots, substitute x with -x in the given polynomial:

x^4-2x^3-4x^2+2x+3  becomes x^4+2x^3-4x^2-2x+3

The coefficients are 1, 2, −4, −2, 3

As can be seen, there are 2 changes.

This means that there are 2 or 0 negative real roots.

Therefore, we conclude that:

  • 2  or 0 positive real roots
  • 2  or 0 negative real roots

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Read 2 more answers
. In a study of air-bag effectiveness it was found that in 821 crashes of midsize cars equipped with air bags, 46 of the crashes
jok3333 [9.3K]

Answer:

P(X ≤ 46 | X~B(821, 0.078)) = 0.00885745584

0.00885... < 0.01

The test statistic of 46 is significant

There is sufficient evidence to reject H₀ and accept H₁

Air bags are more effective as protection than safety belts

Step-by-step explanation:

821 crashes

46 hospitalisations where car has air bags

7.8% or 0.078 probability of hospitalisations in cars with automatic safety belts

α = 0.01 or 1% ← level of significance

One-tailed test

We are testing whether hospitalisations in cars with air bags are less likely than in a car with automatic safety belts;

The likelihood of hospitalisation in a car with automatic safety belts, we are told, is 7.8% or 0.078;

So we are testing if hospitalisations in cars with air bags is less than 0.078;

So, firstly:

Let X be the continuous random variable, the number of hospitalisations from a car crash with equipped air bags

X~B(821, 0.078)

Null hypothesis (H₀): p = 0.078

Alternative hypothesis (H₁): p < 0.078

According to the information, we reject H₀ if:

P(X ≤ 46 | X~B(821, 0.078)) < 0.01

To find P(X ≤ 46) or equally P(X < 47), it could be quite long-winded to do manually for this particular scenario;

If you are interested, the manual process involves using the formula for every value of x up to and including 46, i.e. x = 0, x = 1, x = 2, etc. until x = 46, the formula is:

P(X = r) = nCr * p^{r}  * (1 - p)^{n - r}

You can find binomial distribution calculators online, where you input n (i.e. the number of trials or 821 in this case), probability (i.e. 0.078) and the test statistic (i.e. 46), it does it all for you, which gives:

P(X ≤ 46 | X~B(821, 0.078)) = 0.00885745584

Now, we need to consider if the condition for rejecting H₀ is met and recognise that:

0.00885... < 0.01

There is sufficient evidence to reject H₀ and accept H₁.

To explain what this means:

The test statistic of 46 is significant according to the 1% significance level, meaning the likelihood that only 46 hospitalisations are seen in car crashes with air bags in the car as compared to the expected number in car crashes with automatic safety belts is very unlikely, less than 1%, to be simply down to chance;

In other words, there is 99%+ probability that the lower number of hospitalisations in car crashes with air bags is due to some reason, such as air bags being more effective as a protective implement than the safety belts in car crashes.

5 0
2 years ago
I need help with this​
Kruka [31]

Answer:

X=9/4

Step-by-step explanation:

-3x+10=5x-8

Subtract 10 from each sides

-3x=5x-18

Subtract 5x from both sides

-8x=-18

Divide both sides by -8

x=9/4

I hope this helped

3 0
2 years ago
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