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In-s [12.5K]
3 years ago
14

Write a polynomial f(x) that satisfies the given conditions.

Mathematics
1 answer:
patriot [66]3 years ago
6 0

Answer:

f(x) = (x-4)(x-\frac{1}{2})^2 = x^3-5x^2+4.25x -1

Step-by-step explanation:

Recall, if we have a polynomial of the form (x-a)^k \cdot (x-b)^m, then we say that a is a zero of multiplicity k and b is a zero of multiplicty m. For example, in the polynomial of the form (x+5)^10(x-2)^3 -5 is a zero of multiplicity 10 and 2 is a zero of multiplicity 3. If we want to know the degree of the polynomial,  just add the multiplicity of both zeros (13 in our example).

In this case, we know that the degree of our polynomial should be at least 3(multiplicity 2 and multiplicity 1). So, lets take the polynomial of the form

f(x)=(x-a)(x-b)^2).

In here, a is a zero with multiplicity 1 and b is a zero with multiplicity 2. We are also given that

f(0) = -1 = (0-a)(0-b)^2 = -a\codt b^2.

Which implies that 1 = a\cdot b^2. Since the square of any number is a positive number, it must happen that a>0. So, we have that

b= \pm \sqrt[]{\frac{1}{a}}.

We can choose any value of a and solve for b. Let us choose a=4. So we can have b=1/2 or b=-1/2. Let's use b=1/2. So our polynomial would be

f(x) = (x-4)(x-\frac{1}{2})^2 = x^3-5x^2+4.25x -1

which we can easily check that f(0)=-1.

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Consider the first four terms of an arithmetic sequence below.
VMariaS [17]
7, 13, 19 and 25 have a common difference:  6.  
6 added to 7 gives us 13; 6 added to 13 gives us 19, and so on.

Explicit formula:  a(n) = 7 + 6(n-1), where 7 is the first term and n is the counter (1, 2, 3, ...).

The first term is 7 (given).  This corresponds to n=1.

The second term is a(2) = 7 + 6(2-1), or 7 + 6, or 13.  This corresponds to n = 2.

and so on.


6 0
3 years ago
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Naddika [18.5K]
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5 0
3 years ago
Read 2 more answers
Researchers investigated the possible beneficial effect on heart health of drinking black tea and whether adding milk to the tea
Elanso [62]

Answer:

(a)  It is an experiment.

(b) Group III has been tested to compare the effect of tea

Step-by-step explanation:

24 volunteers are randomly selected to one of the three groups.

Group I drinks two cups of hot black tea without milk

Group II drinks two cups of hot black tea with milk

Group III drinks two cups of hot water but no tea.

At the end of the month, the researchers measured the change in each of the participant's heart health.

The average change of health status of three Groups can be measured and let it be X1', X2' and X3'. We can set up the hypothesis of no difference of heart health i.e H0 : µ1 = µ2 = µ3 against the alternative hypothesis that they are not equal. As the sample size is less than 30, we can use t-statistic.

Under the above logic, (a) it is an experiment.

(b) Group III has been tested to compare the effect of tea or to have comparison between GroupI and Group III.

4 0
3 years ago
A cutting tool fails due to flank wear while machining. It is observed that the tool life at a cutting speed of 160 m/min is 5 m
Ahat [919]

Answer:

The answer is:

30 minutes.

Step-by-step explanation:

We are given the following information:

at 160 m/min, tool life = 5 min.

at 120 m/min, tool life = 17 min.

This means that as the speed reduced from 160 m/min to 120 m/min, the tool life increased from 5 min. to 17 min, hence the difference between the changes are:

speed = 160 - 120 = 40 m/min

tool life = 17 - 5 = 12 min.

Therefore, it can be concluded that a change of speed of 40 m/min, increases the tool life by 12 min.

40 m/min = 12 min

∴ 1 m/min = 12/40 min.

∴ 100 m/min = 12/40 × 100 = 0.3 × 100 =30 min.

8 0
3 years ago
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Answer:

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Step-by-step explanation:

8 0
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