Tutors Answer Your Questions about Polynomialsandrationalexpressions (FREE)
Question 994806: Please help me and show step by step.
1.The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=4 and roots of multiplicity 1 at x=0 and x=−3 . It goes through the point (5,32) .
Find a formula for P(x) .
Answer by rothauserc(2272) (Show Source):
You can put this solution on YOUR website! Root x=4 of multiplicity 2 means that P(x) has a factor (x4)^2
Roots of multiplicity 1 at x=0 means P(x) has a factor x and at
x=3 means a factor (x+3)
****************************************************************
P(x) = x * (x4)^2 * (x+3)
P(x) = (x^2+3x) * (x4)^2
P(x) = (x^2+3x) * (x^2 8x +16)
P(x) = x^4 5x^3 8x^2 +48x
****************************************************************
now we use point (5,32)
32 = 5^4 5(5^3) 8(5^2) +48(5)
32 = 625 625 200 +240
to make this work we have to subtract 8 from the right side of =, therefore
****************************************************************
P(x) = x^4 5x^3 8x^2 +48x 8 and then
32 = 32
Question 994719: expand(y+3r)^3
Answer by Alan3354(47455) (Show Source):
You can put this solution on YOUR website! expand(y+3r)^3

Use Pascal's triangle > coeff's are 1 3 3 1

(y+3r)^3 = 1*y^3 + 3*y^2*(3r)^1 + 3*y*(3r)^2 + 1*(3r)^3
=
Question 994571: simplify (x+y)(x^2xy+y^2)
Answer by stanbon(69061) (Show Source):
Question 994116: Factorize 5z^2_15z_21
Answer by ikleyn(988) (Show Source):
Question 993226: two students took an exam one of them achieved 9 marks more than the other and his marks were 56% of the sum of their marks
Answer by anand429(129) (Show Source):
Question 993883: how do you sketch the graph of a polynomial function that has 3 real zeros, and two imaginary zeros. Like how would you graph this polynomial function 3x^54x^4+2x^3x^2x7. It has 3 real zeros and 2 imaginary,but I don't know how to graph that.
Answer by Theo(5548) (Show Source):
You can put this solution on YOUR website! you have to find the roots.
the graph will cross or touch the xaxis at the real roots.
the graph will not cross or touch the xaxis at the imaginary roots.
if the root has an even multiplicity, the graph will touch the xaxis but not cross it.
if the root has an odd multiplicity,the graph will cross the xaxis.
in between the real roots, you just have to test selected points in order to be able to sketch the graph.
check this reference out as it explains how to sketch a graph and it also discusses even roots and odd roots.
the graph of your equation looks like this:
since it only crosses the xaxis once, it has only 1 real root.
the other 4 must be complex.
fyi  complex roots always come in pairs.
you can have 2 or 4, but you can't have 3.
here's a good reference that should help you understand better.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/
lots of good stuff in there.
check out tutorials 38 and 39.
they apply to your question.
basically you are looking for the real roots.
there are all kinds of tests to determine what they are.
once you found those, if the number of roots is not equal to the degree of the equation, then the other roots must be complex.
Question 993841: Two landscapes must mow a rectangular lawn that measures 100 feet by 200 feet. Each wants to mow no more than half of the lawn. The first starts by mowing around the outside of the lawn. How wide a strip must the first landscaper mow on each side of the four sides in order to mow no more than half of the lawn? The mower has a 24 inch cut. Approximate the required number of trips around the lawn.
I have no idea how to set up an equation for this word problem.
Answer by josgarithmetic(13975) (Show Source):
Question 993820:
Answer by ikleyn(988) (Show Source):
Question 993818: Hi,
I would like to say thank you for your help in advanvced.
Can you please give an explanation of why..
ax + bx 2bx factors into
x(ab)
Please show steps
Thank you,
Anthony
Answer by rothauserc(2272) (Show Source):
Question 993744: howhow do I multiply a number by 2
Answer by Alan3354(47455) (Show Source):
Question 993637: write a polynomial function of least degree with integral coefficients that have the given zero 3i
Answer by stanbon(69061) (Show Source):
You can put this solution on YOUR website! write a polynomial function of least degree with integral coefficients that have the given zero 3i

If the coefficients are integral and 3i is a zero, 3i is also a zero.

Ans: y = (x3i)(x+3i)

y = (x^2  (3i)^2)]

y = x^2 + 9

Cheers,
Stan H.
Question 993574: Please help me solve this problem. It looks simple but I have no idea where to start or what to do?
Solve: x+6/x=5
Found 2 solutions by solver91311, ikleyn: Answer by solver91311(20879) (Show Source): Answer by ikleyn(988) (Show Source):
Question 993493: I am a 7th grade prealgebra student transffering up to take algebra with the 8th graders. I have been trying this problem for about 30 minutes now and I still dont get it. Can you please help out
a triangle has a rational area of 20 ft. what possible combinations or irrational and rational for the base and height would give a rational value of the area.
Answer by Boreal(1464) (Show Source):
You can put this solution on YOUR website! Area is (1/2) bh=20 sq ft. Units will be feet for base and height.
bh=40
h=40/b
b=40/h
As I interpret the question, anything on the below graph where both are positive will give a rational value for the area. If they want possible combinations, you can use anything on this line, like 8 and 5 or 5 and 8 (all are interchangeable), (40/9) and 9, 20/9 and 18, sqrt (40) and sqrt (40), two irrational numbers, sqrt 80 and sqrt (5).
Question 993243: A skydriver jump out of a plane traveling 10,000ft above the ground how high is she after 10 seconds? After 15 seconds
Answer by macston(4006) (Show Source):
You can put this solution on YOUR website! t=time falling, in seconds
.
Distance fallen=(1/2)(acceleration of gravity)(time in seconds)^2
Distance=1/2(32ft/s^2)(t^2)
For 10 seconds:
Distance fallen=1/2(32ft/s^2)(10s)^2=(16ft/s^2)(100s^2)=1600 feet
Height=10000ft1600ft=8400 feet
ANSWER 1: She is 8400 feet high after 10 seconds.
For 15 seconds:
Distance fallen=1/2(32ft/s^2)(15s)^2=(16ft/s^2)(225s^2)=3600 feet
Height=10000ft3600ft=6400 feet
ANSWER 2: She is 6400 feet high after 15 seconds.
Question 993115: = ?
Answer by ikleyn(988) (Show Source):
Question 993057: express in the form a+bi:
1+2i/ 1 square root of 64
Answer by Edwin McCravy(13211) (Show Source):
Question 993065: 2x^2+16x+2 factor using AC method
Answer by Edwin McCravy(13211) (Show Source):
You can put this solution on YOUR website! 2x²+16x+2
Don't start out trying to use the AC method first.
Always look for a common factor first. The AC method
will wait till later  if it can be used at all.
First factor out the common factor 2:
2(x²+8x+1)
As it turns out it is not factorable any further. So
the AC method will not work here. So the final factored
form is simply
2(x²+8x+1)
Edwin
Question 992993: please help me with Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: 4, 1+i
First change the signs on (x(1+i) and (x+(1+i).
giving me (x+1+i) and (x1+i)
I then disturbed and is left with (x^2+2xi2)
Then i disturbed the (x4): (x^3+2ix^22x4x^28xi+8)
That is when i get confused.
,
Answer by ikleyn(988) (Show Source):
You can put this solution on YOUR website! .
Wrong.
Your polynomial should be with real coefficients.
Then, having the root 1+i, it must have the conjugated complex number 1i as the other root.
So the roots must be 4, 1+i and 1i.
Hence, the polynomial is (x4)*(x(1+i))*(x(1i)).
You can transform it further, if you want.
Question 992868: Find polynomials q and r such that f(x)=q(x)g(x)+r(x) where the degree of r is strictly less than degree of g.
f(x)=x^43x^3+2x^2x+1 and g(x)=x^23
Answer by stanbon(69061) (Show Source):
You can put this solution on YOUR website! Find polynomials q and r such that f(x)=q(x)g(x)+r(x) where the degree of r is strictly less than degree of g.
f(x)=x^43x^3+2x^2x+1 and g(x)=x^23

Divid
f(x) by g(x) to find the quotient(q(x)) and the remainder(r(x)).

Ans:
q(x) = x^2  3x +8
r(x) = x + 25

Cheers,
Stan H.

Question 992839: How would I solve this problem? im supposed to divide a rational expression.
Q^2÷2q/6q ÷ q^24/3q^2
Answer by Alan3354(47455) (Show Source):
You can put this solution on YOUR website! How would I solve this problem? im supposed to divide a rational expression.
Q^2÷2q/6q ÷ q^24/3q^2

Invert the divisor and multiply.
The divisor is the fraction on the right side.
==================
It's not clear, add some parentheses.
Question 992784: Dwayne's garden is triangleshaped with two equal sides and a third side that is 4 ft more than the length of an equal side. If the perimeter is 49 ft, how long is each side?
Which equation represents this?
Found 2 solutions by stanbon, Cromlix: Answer by stanbon(69061) (Show Source):
You can put this solution on YOUR website! Dwayne's garden is triangleshaped with two equal sides and a third side that is 4 ft more than the length of an equal side. If the perimeter is 49 ft, how long is each side?
Which equation represents this?

Sketch a picture of this isosceles triangle: equal sides are "x"; base = (x+4).
Equation:
x + x + x+4 = 49
3x = 45
x = 15 (length of each of the equal sides)
x+4 = 19 (length of the base)

Cheers,
Stan H.

Answer by Cromlix(3061) (Show Source):
You can put this solution on YOUR website! Hi there,
Dwayne's garden is an isosceles triangle.
Two equal sides (and two equal angles)
Length of an equal side = 'x'
Length of other side(s) = x + 4
Perimeter = the 3 sides added together.
49 = x + x + (x + 4)
49 = x + x + x + 4
Collect like terms
3x = 49  4
3x = 45
x = 15.
Two equal sides = 15 ft
Third side = 19 ft
Hope this helps :)
Question 992620: Find a polynomial function that has the given zeros
9,5
Answer by MathLover1(11324) (Show Source):
Question 992558: Find the domain and range and sketch the graph for the functions f and g where f(x) is given by the polynomial below and g(x)=f(x). Also use the graph to solve the inequality f(x)<0.
f(x)=9+3x2x^2
Answer by ikleyn(988) (Show Source):
Question 992561: Plot the graph of the rational function f. Include the asymptotes, x and yintercepts and at least one intermediate point between these points.
f(x)=x−2+1/x−1
Answer by stanbon(69061) (Show Source):
You can put this solution on YOUR website! Plot the graph of the rational function f. Include the asymptotes, x and yintercepts and at least one intermediate point between these points.
f(x)= (x−2) + [1/(x−1)]


Cheers,
Stan H.

Question 992556: Write the polynomial as a product of linear factors. Hint: The polynomial has at least one rational zero.
p(x)=x^3+2x^2+3x+6
Answer by stanbon(69061) (Show Source):
You can put this solution on YOUR website! Write the polynomial as a product of linear factors. Hint: The polynomial has at least one rational zero.
p(x)=x^3+2x^2+3x+6

P(x) = x^2(x+2)+3(x+2)

P(x) = (x+2)(x^2+3)

P(x) = (x+2)(x+isqrt(3))(xisqrt(3))

Cheers,
Stan H.

Question 992537: How does this process change if the coefficient of the quadratic term is not either 1 or can be made to be 1 by removing the GCF? For example, consider a quadratic expression such as 14x2 + 29x  15.
Answer by Theo(5548) (Show Source):
You can put this solution on YOUR website! equation is 14x^2 + 29x  15 = 0
multiply the coefficient of the x^2 term by the constant term to get:
14 * 15 = 210
look for factors of 210 that will be equal to 210 when multiplied together and will be equal to 29 when added together.
since the constant term is negative, the two factors will have opposite signs because a positive times a negative is equal to a negative.
looking at all the factors of 210, i find that 35 and 6 are the factors that i need.
this is because 35 * 6 = 210 and 35  6 = 29.
the positive factor is 35.
the negative factor is 6.
now you split the middle term of the equation as shown below:
14x^2 + 29x  15 = 0 becomes:
14x^2 + 35x  6x  15 = 0
now group the first two terms and the last two terms together to get:
(14x^2 + 35x)  (6x + 15) = 0
note the change of sign in the second of these grouped factors.
grouping with minus sings can be tricky.
 6x  15 becomes  (6x + 15) when you group those last two factors together.
that's because  (6x + 15) is equivalent to 6x  15 when you expand it.
now factor out the common terms of reach group to get:
7x * (2x + 5)  3 * (2x + 5)
what you are looking for here is that one of the factors in the secondf set is common to one of the factors in the first set.
the common factor that we were looking for here is 2x + 5
now you can factor out that common factor to get:
7x * (2x + 5)  3 * (2x + 5) becomes:
(7x  3) * (2x + 5)
those are your factors.
multiply them out and you will see that you get the original equation.
(7x  3) * (2x + 5) =
7x * 2x = 14x^2
+ 7x * 5 = 35x
 3 * 2x = 6x
 3 * 5 = 15
combine like terms and you get:
14x^2 + 29x  15.
that's your original equation so you're done.
this method is tricky but gets you to the right answer fairly quickly if you do it right.
same with the box method.
the other method is guess and check where you go through a ritual of trying dfifferent factors until you arrive at the right one.
if any of these methods prove difficult, then go to the quadratic formula.
that will be you the factors in all case, whether or not the quadratic equation is factorable or not, and whether or not the factors are real or not.
here's some references on factoring that you might find helpful.
http://www.purplemath.com/modules/solvquad.htm
http://www.purplemath.com/modules/factquad.htm
http://www.regentsprep.org/regents/math/algtrig/atv1/revfactorgrouping.htm
https://www.youtube.com/watch?v=Od38CRJNC5w
all kinds of stuff on the web.
all you need to do is a search on "factoring quadratic equations" or "factoring quadratics using box method" or "factoring quadratic equations using split the middle term method.
there are even other methods like the indian method.
personally, if i can't factor it easily using any of these methods, i go right to the quadratic formula since that is the method of last resort.
Question 992529: If m= 3 then what is the value of 2M
Answer by Timnewman(249) (Show Source):
Question 992351: I need to find the real and complex solution(s) and whether it has a multiplicity.
Problem: f(x) = x^47x^2+ 12
Note: I am able to solve similar equations if the exponents are sequential. However, I am unsure how to solve when they are not. Example...x^4  X^3 + 25x^2  25x = Solutions are x =1, 5i, and 5i.
Many Thanks!
Found 3 solutions by MathTherapy, ikleyn, stanbon: Answer by MathTherapy(4047) (Show Source): Answer by ikleyn(988) (Show Source): Answer by stanbon(69061) (Show Source):
You can put this solution on YOUR website! I need to find the real and complex solution(s) and whether it has a multiplicity.
Problem: f(x) = x^47x^2+ 12
Factor::
(x^26)(x^21) = 0
x^2 = 6 or x^2 = 1

x = +/sqrt(6) or x = +/1

Note: I am able to solve similar equations if the exponents are sequential. However, I am unsure how to solve when they are not.
Example...x^4  X^3 + 25x^2  25x = Solutions are x =1, 5i, and 5i.

Rearrange::
x^4+25x^2  (x^3+25x) = 0
x^2(x^2+25)  x(x^2+25) = 0

(x^2x)(x^2+25 = 0

x(x1)(x5i)(x+5i) = 0
x = 0 or x = 1 or x = 5i or x = 5i

Cheers,
Stan H.

Question 992121: How do I simplify this expression: (x8)/(x4)+(x5)/(x7)2
using this method of simplifying: (a4)/(a5)(a5)/(a6)= [1+ 1/(a5)][1+ 1/(a6)= 1/(a5)1/(a6)= 1/[(a5)(a6)]
Thank you
Found 2 solutions by MathTherapy, josgarithmetic: Answer by MathTherapy(4047) (Show Source):
You can put this solution on YOUR website!
How do I simplify this expression: (x8)/(x4)+(x5)/(x7)2
using this method of simplifying: (a4)/(a5)(a5)/(a6)= [1+ 1/(a5)][1+ 1/(a6)= 1/(a5)1/(a6)= 1/[(a5)(a6)]
Thank you
Simplifies to: , or
Answer by josgarithmetic(13975) (Show Source):
Question 991924: Please help to simplify (1/x+1/y)/(1/x1/y)
Please show detailed workings thank you.
Found 2 solutions by ikleyn, josgarithmetic: Answer by ikleyn(988) (Show Source): Answer by josgarithmetic(13975) (Show Source):
Question 991597: rational expression to work on and factor polynomials completely:
81b^216
_________
59
Answer by MathLover1(11324) (Show Source):
Question 991560: sum of the squares of two numbers is 162. the product of the two numbers is 181. Find the numbers
Answer by Fombitz(25151) (Show Source):
Question 991423: is pi a polynomial?
Answer by Fombitz(25151) (Show Source):
Question 991403: What is the factored form of x squared minus x minus 6
Answer by Alan3354(47455) (Show Source):
Question 989965: if 3x+2y=12 and xy=6 then what is 27x^3+8y^3
Answer by Fombitz(25151) (Show Source):
Question 991360: Factor the following polynomial completely.
(y + 4)^2 –2(y + 4) + 1
Found 3 solutions by MathTherapy, josgarithmetic, Fombitz: Answer by MathTherapy(4047) (Show Source): Answer by josgarithmetic(13975) (Show Source): Answer by Fombitz(25151) (Show Source):
Question 991132: How to find the hcf of 30,90 and 180
Answer by Fombitz(25151) (Show Source):

Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 3871..3915, 3916..3960, 3961..4005, 4006..4050, 4051..4095, 4096..4140, 4141..4185, 4186..4230, 4231..4275, 4276..4320, 4321..4365, 4366..4410, 4411..4455, 4456..4500, 4501..4545, 4546..4590, 4591..4635, 4636..4680, 4681..4725, 4726..4770, 4771..4815, 4816..4860, 4861..4905, 4906..4950, 4951..4995, 4996..5040, 5041..5085, 5086..5130, 5131..5175, 5176..5220, 5221..5265, 5266..5310, 5311..5355, 5356..5400, 5401..5445, 5446..5490, 5491..5535, 5536..5580, 5581..5625, 5626..5670, 5671..5715, 5716..5760, 5761..5805, 5806..5850, 5851..5895, 5896..5940, 5941..5985, 5986..6030, 6031..6075, 6076..6120, 6121..6165, 6166..6210, 6211..6255, 6256..6300, 6301..6345, 6346..6390, 6391..6435, 6436..6480, 6481..6525, 6526..6570, 6571..6615, 6616..6660, 6661..6705, 6706..6750, 6751..6795, 6796..6840, 6841..6885, 6886..6930, 6931..6975, 6976..7020, 7021..7065, 7066..7110, 7111..7155, 7156..7200, 7201..7245, 7246..7290, 7291..7335, 7336..7380, 7381..7425, 7426..7470, 7471..7515, 7516..7560, 7561..7605, 7606..7650, 7651..7695, 7696..7740, 7741..7785, 7786..7830, 7831..7875, 7876..7920, 7921..7965, 7966..8010, 8011..8055, 8056..8100, 8101..8145, 8146..8190, 8191..8235, 8236..8280, 8281..8325, 8326..8370, 8371..8415, 8416..8460, 8461..8505, 8506..8550, 8551..8595, 8596..8640, 8641..8685, 8686..8730, 8731..8775, 8776..8820, 8821..8865, 8866..8910, 8911..8955, 8956..9000, 9001..9045, 9046..9090, 9091..9135, 9136..9180, 9181..9225, 9226..9270, 9271..9315, 9316..9360, 9361..9405, 9406..9450, 9451..9495, 9496..9540, 9541..9585, 9586..9630, 9631..9675, 9676..9720, 9721..9765, 9766..9810, 9811..9855, 9856..9900, 9901..9945, 9946..9990, 9991..10035, 10036..10080, 10081..10125, 10126..10170, 10171..10215, 10216..10260, 10261..10305, 10306..10350, 10351..10395, 10396..10440, 10441..10485, 10486..10530, 10531..10575, 10576..10620, 10621..10665, 10666..10710, 10711..10755, 10756..10800, 10801..10845, 10846..10890, 10891..10935, 10936..10980, 10981..11025, 11026..11070, 11071..11115, 11116..11160, 11161..11205, 11206..11250, 11251..11295, 11296..11340, 11341..11385, 11386..11430, 11431..11475, 11476..11520, 11521..11565, 11566..11610, 11611..11655, 11656..11700, 11701..11745, 11746..11790, 11791..11835, 11836..11880, 11881..11925, 11926..11970, 11971..12015, 12016..12060, 12061..12105, 12106..12150, 12151..12195, 12196..12240, 12241..12285, 12286..12330, 12331..12375, 12376..12420, 12421..12465, 12466..12510, 12511..12555, 12556..12600, 12601..12645, 12646..12690, 12691..12735, 12736..12780, 12781..12825, 12826..12870, 12871..12915, 12916..12960, 12961..13005, 13006..13050, 13051..13095, 13096..13140, 13141..13185, 13186..13230, 13231..13275, 13276..13320, 13321..13365, 13366..13410, 13411..13455, 13456..13500, 13501..13545, 13546..13590, 13591..13635, 13636..13680, 13681..13725, 13726..13770, 13771..13815, 13816..13860, 13861..13905, 13906..13950, 13951..13995, 13996..14040, 14041..14085, 14086..14130, 14131..14175, 14176..14220, 14221..14265, 14266..14310, 14311..14355, 14356..14400, 14401..14445, 14446..14490, 14491..14535, 14536..14580, 14581..14625, 14626..14670, 14671..14715, 14716..14760, 14761..14805, 14806..14850, 14851..14895, 14896..14940, 14941..14985, 14986..15030, 15031..15075, 15076..15120, 15121..15165, 15166..15210, 15211..15255, 15256..15300, 15301..15345, 15346..15390, 15391..15435, 15436..15480, 15481..15525, 15526..15570, 15571..15615, 15616..15660, 15661..15705, 15706..15750, 15751..15795, 15796..15840, 15841..15885, 15886..15930, 15931..15975, 15976..16020, 16021..16065, 16066..16110, 16111..16155, 16156..16200, 16201..16245, 16246..16290, 16291..16335, 16336..16380, 16381..16425, 16426..16470, 16471..16515, 16516..16560, 16561..16605, 16606..16650, 16651..16695, 16696..16740, 16741..16785, 16786..16830, 16831..16875, 16876..16920, 16921..16965, 16966..17010, 17011..17055, 17056..17100, 17101..17145, 17146..17190, 17191..17235, 17236..17280, 17281..17325, 17326..17370, 17371..17415, 17416..17460, 17461..17505, 17506..17550, 17551..17595, 17596..17640, 17641..17685, 17686..17730, 17731..17775, 17776..17820, 17821..17865, 17866..17910, 17911..17955, 17956..18000, 18001..18045, 18046..18090, 18091..18135, 18136..18180, 18181..18225, 18226..18270, 18271..18315, 18316..18360, 18361..18405, 18406..18450, 18451..18495, 18496..18540, 18541..18585, 18586..18630, 18631..18675, 18676..18720, 18721..18765, 18766..18810, 18811..18855, 18856..18900, 18901..18945, 18946..18990, 18991..19035, 19036..19080, 19081..19125, 19126..19170, 19171..19215, 19216..19260, 19261..19305, 19306..19350, 19351..19395, 19396..19440, 19441..19485, 19486..19530, 19531..19575, 19576..19620, 19621..19665, 19666..19710, 19711..19755, 19756..19800, 19801..19845, 19846..19890, 19891..19935, 19936..19980, 19981..20025, 20026..20070, 20071..20115, 20116..20160, 20161..20205, 20206..20250, 20251..20295, 20296..20340, 20341..20385, 20386..20430, 20431..20475, 20476..20520, 20521..20565, 20566..20610, 20611..20655, 20656..20700, 20701..20745, 20746..20790, 20791..20835, 20836..20880, 20881..20925, 20926..20970, 20971..21015, 21016..21060, 21061..21105, 21106..21150, 21151..21195, 21196..21240, 21241..21285, 21286..21330, 21331..21375, 21376..21420, 21421..21465, 21466..21510, 21511..21555, 21556..21600, 21601..21645, 21646..21690, 21691..21735, 21736..21780, 21781..21825, 21826..21870, 21871..21915, 21916..21960, 21961..22005, 22006..22050, 22051..22095, 22096..22140, 22141..22185, 22186..22230, 22231..22275, 22276..22320, 22321..22365, 22366..22410, 22411..22455, 22456..22500, 22501..22545, 22546..22590, 22591..22635, 22636..22680, 22681..22725, 22726..22770, 22771..22815, 22816..22860, 22861..22905, 22906..22950, 22951..22995, 22996..23040, 23041..23085, 23086..23130, 23131..23175, 23176..23220, 23221..23265, 23266..23310, 23311..23355, 23356..23400, 23401..23445, 23446..23490, 23491..23535, 23536..23580, 23581..23625, 23626..23670, 23671..23715, 23716..23760, 23761..23805, 23806..23850, 23851..23895, 23896..23940, 23941..23985, 23986..24030, 24031..24075, 24076..24120, 24121..24165, 24166..24210, 24211..24255, 24256..24300, 24301..24345, 24346..24390, 24391..24435, 24436..24480, 24481..24525, 24526..24570, 24571..24615, 24616..24660, 24661..24705, 24706..24750, 24751..24795, 24796..24840, 24841..24885, 24886..24930, 24931..24975, 24976..25020, 25021..25065, 25066..25110, 25111..25155, 25156..25200, 25201..25245, 25246..25290, 25291..25335, 25336..25380, 25381..25425, 25426..25470, 25471..25515, 25516..25560, 25561..25605, 25606..25650, 25651..25695, 25696..25740, 25741..25785, 25786..25830, 25831..25875, 25876..25920, 25921..25965, 25966..26010, 26011..26055, 26056..26100, 26101..26145, 26146..26190, 26191..26235, 26236..26280, 26281..26325, 26326..26370, 26371..26415, 26416..26460, 26461..26505, 26506..26550, 26551..26595, 26596..26640, 26641..26685, 26686..26730, 26731..26775, 26776..26820, 26821..26865, 26866..26910, 26911..26955, 26956..27000, 27001..27045, 27046..27090, 27091..27135, 27136..27180, 27181..27225, 27226..27270, 27271..27315, 27316..27360, 27361..27405, 27406..27450, 27451..27495, 27496..27540, 27541..27585, 27586..27630, 27631..27675, 27676..27720, 27721..27765, 27766..27810, 27811..27855, 27856..27900, 27901..27945, 27946..27990, 27991..28035, 28036..28080, 28081..28125, 28126..28170, 28171..28215, 28216..28260, 28261..28305, 28306..28350, 28351..28395, 28396..28440, 28441..28485, 28486..28530, 28531..28575, 28576..28620, 28621..28665, 28666..28710, 28711..28755, 28756..28800, 28801..28845, 28846..28890, 28891..28935, 28936..28980, 28981..29025, 29026..29070, 29071..29115, 29116..29160, 29161..29205, 29206..29250, 29251..29295, 29296..29340, 29341..29385, 29386..29430, 29431..29475, 29476..29520, 29521..29565, 29566..29610, 29611..29655, 29656..29700, 29701..29745, 29746..29790, 29791..29835, 29836..29880, 29881..29925, 29926..29970, 29971..30015, 30016..30060, 30061..30105, 30106..30150, 30151..30195, 30196..30240, 30241..30285, 30286..30330, 30331..30375, 30376..30420, 30421..30465, 30466..30510, 30511..30555, 30556..30600, 30601..30645, 30646..30690, 30691..30735, 30736..30780, 30781..30825, 30826..30870, 30871..30915, 30916..30960, 30961..31005, 31006..31050, 31051..31095, 31096..31140, 31141..31185, 31186..31230, 31231..31275, 31276..31320, 31321..31365, 31366..31410, 31411..31455, 31456..31500, 31501..31545, 31546..31590, 31591..31635, 31636..31680, 31681..31725, 31726..31770, 31771..31815, 31816..31860, 31861..31905, 31906..31950, 31951..31995, 31996..32040, 32041..32085, 32086..32130, 32131..32175, 32176..32220, 32221..32265, 32266..32310, 32311..32355, 32356..32400, 32401..32445, 32446..32490, 32491..32535, 32536..32580, 32581..32625, 32626..32670, 32671..32715, 32716..32760, 32761..32805, 32806..32850, 32851..32895, 32896..32940, 32941..32985, 32986..33030, 33031..33075, 33076..33120, 33121..33165, 33166..33210, 33211..33255, 33256..33300, 33301..33345, 33346..33390, 33391..33435, 33436..33480, 33481..33525, 33526..33570, 33571..33615, 33616..33660, 33661..33705, 33706..33750, 33751..33795, 33796..33840, 33841..33885, 33886..33930, 33931..33975, 33976..34020
