What is the factor of x^2-11+24?
450981
I'm guessing that second term was to be -11x.\n\nSadly, this is trial and error.  as the last term (+24) you know the terms are both like signed (both x+something or X-something).  The second term being negative (-11x), we know that they are both x-something.  so we are looking for two factors:  (x-a) and (x-b), where:\n\n1.  a+b=11, and\n2.  axb=24\n\nOur factors of 24 are:\n1x24\n2x12\n3x8\n4x6\n\nas 3x8 fits, we try it:\n\n(x-3) (x-8)\n\nthis equals:\n\nx^2-11x+24\n\nso we are right with these two factors\n\nHope this helps....I do these things instinctually, so I have trouble explaining them.
809123
(x-2)(x+1)(x-1)-6(x+1)\nNow common factor is x+1\n(x+1)[(x-2)(x-1)-6)]\n(x+1)(x^2-3x+2-6)\n(x+1)(x^2-3x-4)\n(x+1)(x-4)(x+1)\n(x-4)(x+1)^2
561923
You need to determine 2 numbers that add to -11 (the x term) and multiply to -60 (the numerical term).  You can do it by formula, or by a little trial-and-error.\n\nThe answers are -15 and 4, so the equation factors:\nx^2 - 11x - 60 = 0\n(x - 15)(x + 4) = 0\n\nIn order for the total to be zero, at least one of the factors must be zero.  So your solutions are x = 15 (which makes the first factor zero) and x = -4 (which makes the second factor zero).
226307
= 24 k^2   +  6 k^3 +  18k^2  - 54 k^3\n= 42k^2 - 48 k^3\n= 6k^2 (7 - 8k)\n\nif 6k^2 (7 - 8k)=0\n7 - 8k=0\n8k=7\nk=7/8
853717
The squares of two of the numbers must add to the square of the third.\nLets try x^2 = 24^2 + 51^2, then x^2 = 3177. Thats not a square.\nLets try 51^2 = 24^2 + x^2. Then x^2 = 2025, so x = 45.\nSo (24,45,51) is a pythagorean triple.
879678
The first thing to do is to find the factors of the number you're taking the root of.\nFor example, teh factors of 24 include 2*12, 3*8, and 4*6.\nSince four is a perfect square (2^2), we'll use that set of factors.\nSo:\nsqrt(24)= sqrt(4*6)= sqrt(4)*sqrt(6)= 2sqrt(6)\n\nSame with sqrt(18):\n18=9*2\n9=3^2\nsqrt(18)= sqrt(9)*sqrt(2)= 3sqrt(2)\n\nsqrt(3) +sqrt(3) can only be reduced by realizing that it's\n2*sqrt(3), which could be the sqrt(12). THis is because you can take the two under the radical by squaring it. so  you get sqrt (2^2+3)= sqrt(12), but technically 2sqrt(3) is more reduced.\n\nsqrt(27)=3 sqrt(3)\nsqrt(12)= 2 sqrt(3)\nso 3sqrt(3) + 2sqrt(3) = 5sqrt(3)\nin this case, you're treating the sqrt(3) sort of like it was a variable. think of 3x +2x = 5x
979750
Because you know that 1 - 11i is a zero, then 1 + 11i is also a zero, so \n\n( x - ( 1 - 11i ) ) ( x - ( 1 + 11i ) ) is a factor of the expression.\n\nSimplifying, get x^2 - 2x + 122 is a factor.\n\nDivide to find the quotient, using "long division" [See the web page for doing this]:\n\n x^3 - 2x^2 + 9x - 18\n-------------------------------\nx^2-2x+122 ) x^5 -4x^4 + 135x^3 - 280x^2 + 1134x - 2196\nx^5 -2x^4 + 122x^3\n-----------------------\n-2x^4 + 13x^3\n-2x^4 + 4x^3 - 244x^2\n--------------------------\n9x^3 - 36x^2\n9x^3 - 18x^2 + 1098x\n--------------------------\n-18x^2 + 36x\n\nSo, now have to factor x^3 - 2x^2 + 9x - 18. If you did not recognize that it equals x^2 ( x -2 ) + 9 ( x - 2 ) = ( x^2 + 9 ) ( x - 2 ), you can try small numbers: 1, -1, 2, -2, etc. Find that when x = 2\n\n2^3 - 2(2^2) + 9(2) - 18 = 0, so ( x - 2 ) is a factor.\n\nNow divide x^3 - 2x^2 + 9x - 18 by x - 2, again by long division to get\n\nx^2 + 9.\n\nSo the factorization is\n\nf(x) = ( x - ( 1 - 11i) ) ( x - ( 1 + 11i) )  * ( x - 2 ) * ( x^2 + 9 )\n\nx^2 + 9 = 0 has zeros 3i and -3i, so the zeros of f(x) are:\n\n1 - 11i, 1 + 11i, 2, 3i and -3i.
976644
x ^2 - x - 72 =0\nfactor 72\n8*9=72\ntherefore\nx^2 -9x+8x-72=0\nx(x-9)+8(x-9)=0\n(x-9)(x+8)=0\nx=9;-8
774953
It's quite complicated.  The link given goes into detail about an inductive method of obtaining this.  If you just want the answer it is -\n\n(n^4 - 6n^3 + 23n^2 -18n + 24) / 24
779020
Just a number. Also, the answer is 3, not 37. You just have to substitute -1/3 where there is an x in the original.\n\nf(x) = 18x(squared) - 12x - 3\nf(-1/3) = 18 * (-1/3)^2 - 12 * ( -1/3) - 3\n                    18 * 1/9     + 4            - 3\n                             2 + 4 -3\n                                  3
950667
This is a mathematics question, but fine. \nThe factors are 1, 2, 3, 4, 6, 8, 12, 16, 24.\nThe prime factors are 2^4,3.\n\nJust trial and error.
289855
First factor out the x: x(x^2 + 2x - 63)\n\nThen FOIL:  x(x+9)(x-7)
402918
I think you wrote 10x*2yz - 22x*3yz.  To factor this oyu must take out like terms by dividing.  You can see that both terms share a 2, y, z, and an x*2.  After the  division you get  2x*2yz( 5 - 11x).  I hope this helps.
925677
x^n + y^n has factors when n is not a power of 2: n = 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...\n\nIf n is odd then ( x + y ) is a factor, by the factor theorem: put y = -x.\n\nIf n has an odd factor n = p q where p is odd, then x^(pq) = (x^q)^p. So, x^n + y^n = (x^q)^p + (y^q)^p can be factored. Example: x^44 + y^44 = (x^4)^11 + (y^4)^11 has x^4 + y^4 as a factor.\n\nThat only leaves x^(2^n) + y^(2^n): x^2 + y^2, x^4 + y^4, x^8 + y^8, ... None of these can be factored, using Eisenstein's criteria.\n\nIn the above I assume that all coefficients are integers. If not, note that\n\nx^4 + y^4 = ( x² + √2 x + 1 ) ( x² - √2 x + 1 )\n\nand similar results for other powers of 2.
949394
x^2 + 16x + y^2 -22y = 20\n\nComplete the square for x and y:\n\nx^2 + 16x + 64 + y^2 -22y + 121 = 20 + 64 + 121\n\nFactor:\n\n(x+8)^2 + (y-11)^2 = 205\n\nCenter is (-8, 11) and radius = sqrt (205)
136974
2x is the largest possible factor, leaving:\n2x(4x^2-5x-7)
311512
a comon factor is -3x\n-12x^4-3x=\n-3x*4x^3-3x*1=\n-3x(4x^3+1)