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marin [14]
3 years ago
11

Report Error Suppose $P(x)$ is a polynomial of smallest possible degree such that: $\bullet$ $P(x)$ has rational coefficients $\

bullet$ $P(-3) = P(\sqrt 7) = P(1-\sqrt 6) = 0$ $\bullet$ $P(-1) = 8$ Determine the value of $P(0)$.
Mathematics
1 answer:
motikmotik3 years ago
4 0

Answer:

We want a polynomial of smallest degree with rational coefficients with zeros in \sqrt{7}, 1 - \sqrt{6} and -3. The last root gives us the factor (x+3). Hence, our polynomial is

P(x) =(x+3)q(x)

where q is a polynomial with rational coefficients and roots \sqrt{7} and 1 - \sqrt{6}. The root \sqrt{7} gives us a factor x-\sqrt{7}, but in order to obtain rational coefficients we must consider the factor x^2-7.

An analogue idea works with 1 - \sqrt{6}. For convenience write  x - 1 + \sqrt{6} = ( x - 1) + \sqrt{6}. This gives the factor (x-1)^2-6. Hence,

P(x) = (x+3)(x^2-7)((x-1)^2-6)=x^5+x^4-18x^3-22x^2+77x+105

Notice that P(-1)=24. So, in order to satisfy the last condition we divide by 3 the whole polynomial, without altering its roots. Finally, the wanted polynomial is

P(x) =(1/3)x^5+(1/3)x^4-6x^3-(22/3)x^2+(77/3)x+35

Step-by-step explanation:

We must have present that any polynomial it's determined by its roots up to a constant factor. But here we have irrational ones, in order to eliminate the irrational coefficients that a factor of the type x-\sqrt7 will introduce in the expression, we need to multiply by its conjugate x+\sqrt7. Hence, we will obtain x^2-7 that have rational coefficients. Finally, the last condition is given with the intention to fix the constant factor. Usually it is enough to evaluate in the point and obtain the necessary factor.

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The Martinez family has 34 gallon of orange juice in the refrigerator. How many cups of orange juice are in the refrigerator?
kifflom [539]

Answer:

544 cups of orange juice

Step-by-step explanation:

1 gallon = 16 cups

34 x 16 = 544

5 0
2 years ago
Two cities have nearly the same​ north-south line of 110 degrees Upper W. The latitude of the first city is 23 degrees Upper F c
inn [45]

Answer:

CORRECTED QUESTION:

Two cities have nearly the same​ north-south line of 110 degrees Upper W. The latitude of the first city is 23 degrees Upper N, and the latitude of the second city is 36 degrees N. Approximate the distance between the cities if the average radius of Earth is 6400 km.

ANSWER: 1452.11 km

Step-by-step explanation:

Since the two cities both lies on the Northern latitude of the sphere along  the same longitude, we are going to subtract the angles the latitude that each city subtend at the equator.

36 - 23 = 13 degrees   i.e the angles between the with two cities on a cross section the large circle formed by the longitude and its center.

Applying   the formula for the length of an arc on a sector on the large circle  

(∅/ 360) x 2πR

where, ∅ = is the angle between the two cities

R = radius of the Earth.

13/360 x 2 x π x 6400 = 1452.11 km

7 0
3 years ago
What is the answer to, one half of 9?
GREYUIT [131]
Answer:

4.5

Step-by-step explanation:

One half of 9 = 9/2 = 4.5
7 0
2 years ago
Identify the type of sequence shown and select the appropriate response.
Alexus [3.1K]

Answer:

the answer is A

Step-by-step explanation:

if you multiple 6 to 4 you get 24. if you multiple 24 to 6 you get 144. if you multiple 144 to 6 you get 864.

6 0
3 years ago
Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the inter
slava [35]

Using it's concept, the average rate of change of the function over the interval 10 < x < 22 is given by:

r = \frac{f(22) - f(10)}{12}

<h3>What is the average rate of change of a function over an interval?</h3>

It is given by the change in the <u>output divided by the change in input</u>.

Over the interval 10 < x < 22, the outputs are f(10) and f(22), hence the rate of change is given by:

r = \frac{f(22) - f(10)}{12}

More can be learned about the average rate of change of a function at brainly.com/question/24313700

#SPJ1

8 0
2 years ago
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