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1. The remainder is x^2 - 4x + 7.

2. f(x) = 36x^5 - 20x^4 + 5x^3.

Thank you :DD

2. By calculus,

\([\mathcal{R}_1] = 30 + \frac{1}{4} \cdot \pi \cdot 10^2 + \int_0^5 \sqrt{100 - x^2} \ dx + \int_0^6 \sqrt{100 - x^2} \ dx.\)

We can write out the areas similarly, to get [R_1] - [R_2] - [R_3] + [R_4] = 84.

\(T_b = T_s[60]-T_g[21]\)

\(T_b = 39\)

\(J_b= T_b[39]-S_b[20]\)

\(J_b = 19\)

\(J_g = T_j[34]-J_b[19]\)

\(J_g= 15\)

_Answer:

\(15\)

2log(a - 2b)  =  log(a) + log(b)

Using rules of logs:  2log(a - 2b)  =  log(a - 2b)²

                                 log(a) + log(b)  =  log(ab)

Therefore:  log(a - 2b)²  =  log(ab)

--->                  (a - 2b)²  =  ab

              a² - 4ab + 4b²  =  ab

              a² - 5ab + 4b²  =  0

               (a - 4b)(a - b)  =  0

So, either  a - 4b  =  0   --->   a  =  4b   --->   a/b  =  4  

or  a - b  =  0   --->   a  =  b  (but this is impossible for it will create a log of a negative number)

\({-{1\over 2}}= {{1\over 2}\over{1-{1\over 2}}}\)

₌

\(-{3\over 2}\)

(3x - 17)/((x - 3)(x - 7) = A/(x - 3) + B/(x - 7)

Rewrite each fraction with the denominator:  (x - 3)(x - 7) 

[ A/(x - 3) ] · [ (x - 7)/(x - 7) ]  =  [ A · (x - 7) ] / [ (x - 3)(x - 7) ]  =  [ Ax - 7A ] / [ (x - 3)(x - 7) ] 

[ B(x - 7) ] · [ (x - 3)/(x - 3) ]  =  [ B · (x - 3) ] / [ (x - 3)(x - 7) ]  =  [ Bx - 3B ] / [ (x - 3)(x - 7) ] 

Adding these two expressions together:

[ Ax - 7A ] / [ (x - 3)(x - 7) ]  +  [ Bx - 3B ] / [ (x - 3)(x - 7) ]   =  [ Ax - 7A + Bx - 3B ] / [ (x - 3)(x - 7) ] 

Comparing the original numerator  ( 3x - 17 )  to this numerator  ( Ax - 7A + Bx - 3B )

3x - 17  =  Ax - 7A + Bx - 3B

3x - 17  =  (Ax + Bx) + (- 7A - 3B)

Setting the x-terms equal to each other:       3x  =  Ax + Bx   --->     3  =  A + B

Setting the constants equal to each other:  -17  =  -7A - 3B   --->   17  =  7A + 3B

Solving for A:   A + B  =  3   --->   -A -    B  =  -3   --->   -3A - 3B  =  -9

                                                    7A + 3B  =  17   --->   7A + 3B  =  17

Solving:                                                                          4A           =  8     --->     A  =  2

Since  A + B  =  3     --->     2 + B  =  3     --->   B  =  1

i have no clue how you even got the radius...

a = -1 and b = 4, so a^2 + b^2 = 17.

That's the wrong answer.

There are 85 integers that work.

a/b works out to 6.

it is locked, no one can answer, and you are not the questioner in the first place/anyway.

Number 8 appears 20 times between 1 and 100 as follows:

8, 18, 28, 38, 48, 58, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 98 =20 such digits.

You have the following pairs:  [153, 9], [ 17, 17], [ 9, 153]

Your question is vague !

If: [2007 + 1/2007] - [2006  +  1/2006] =0.99999975161709689069314229707489 . Is that what you meant?

The point of intersection is (1,-4,0).

The angle between the line and the plane is 15 degrees.

\(A_Q ={1\over5}AB\)

\(Arc[AP]= {1\over 5}Arc[AB]\)

\(PQ = { \sqrt{(P_O)​​^2-(AB)​​^2}}\)

\(PQ = 4\)

\(PB = {\sqrt{​​(PQ)^2+(QB)^2} }\)

\(PB =\sqrt{80}\)

_R:

\(PB = {4\sqrt{5}}\)

As pointed out, a part of the problem you have raised refers to Armstrong numbers / narcissistic numbers, i.e., nn -digit numbers equal to the sum of nn -th power of its digits. Besides that, there is nothing really special about such numbers. For n=3n=3 , there are only 44 such numbers, 153,370,371153,370,371 and 407407 . There is no analytical means to solve those equations exactly, thus one has to rely on trial-and-error or brute force.

The other part, where a 4-digit number is equal to sum of third powers of its digits, does not have any solution. Consider, as you showed:

a3+b3+c3=100a+10b+ca3+b3+c3=100a+10b+c

Adding, 10001000 on both sides, we have:

103+a3+b3+c3=1000+100a+10b+c103+a3+b3+c3=1000+100a+10b+c

Clearly, for any d∈[1,9]d∈[1,9] , we have:

103+a3+b3+c3≤1000d+100a+10b+c103+a3+b3+c3≤1000d+100a+10b+c

And thus:

d3+a3+b3+c3<1000d+100a+10b+cd3+a3+b3+c3<1000d+100a+10b+c

Conclusion: no solutions exist.

Hope this helps, whymenotsmart^m^, although guest's answer is alternable:)

List the factors of 216.

 - 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216

Since his ID code is only three-digits, we can eliminate the numbers above 9, which leaves us to:

 - 1, 2, 3, 4, 6, 8, 9

Use these numbers to determine what 3 numbers add to equal 19 and multiply to equal 216.

The numbers in Angel's ID code include: 4, 6, and 9.

List the possible three-digit outcomes of 4, 6, 9 (hint - since there are three digits, there are 6 outcomes):

4-6-9

4-9-6

9-4-6

9-6-4

6-9-4

6-4-9

The last bit of information also tells us that the integer code is as large as possible. Picking from the list of possible three-digit outcomes, ask yourself, which is the largest possible?
 

Angelo's ID code is 9-6-4.

If you have any questions, don't be afraid to ask! I hope I explained well enough. 

:)