標題:
about quartic polynomial
發問:
How to express quartic polynomial into sum of 2 square easily? For example, Given 圖片參考：http://i165.photobucket.com/albums/u61/henrywan910/latex-1.png but how we can solve it into 2 square with ease 更新: you maybe right when the other square equal 1 but 2 x^4+18 x^3+41 x^2-18 x+2 how can you solve it into 2 squares?
最佳解答:
(x^2+Ax+B)^2 =x^4+2Ax^3+(A^2+2B)x^2+2ABx+B^2 Comparing coefficient for x^3, 2A=-8 => A=-4 Comparing coefficient for x^2, (A^2+2B)=22 so 16+2B=22 => B=3 So (x^2-4x+3)^2=x^4-8x^3+22x^2-24x+9 ...... 2009-06-14 01:44:51 補充： Expressing a quartic polynomial f(x) into sum of 2 squares is not always possible because it means the quartic polynomial must be positive, so no real root for f(x)=0. So I guess your answer is for specific polynomials only. 2009-06-14 01:45:01 補充： 2 x^4+18 x^3+41 x^2-18 x+2 =(x^2+ax+b)^2+(x^2+cx+d)^2 =2x^4+2(a+c)x^3+(a^2+c^2+2b+2d)x^2+2(ab+cd)x+(b^2+d^2)=0 2009-06-14 01:45:11 補充： Comparing terms, 2(a+c)=18...(1) a^2+c^2+2b+2d=41...(2) 2(ab+cd)=-18...(3) b^2+d^2=2...(4) 2009-06-14 01:45:17 補充： Since it is specific polynomial, I have to make some guess. Looking at (4) I would guess b=+/-1 and d=+/-1 Put these combinations into (3) will always contradicts (1) except when b=-1 and d=-1. Sub into (2) gives a^2+c^2=45 Combine with (1) gives a,c=6,3 or 3,6 So it is (x^2+3x-1)^2 + (x^2+6x-1)^2 2009-06-14 01:47:19 補充： The senstence should read: Put these combinations into (3) will always contradicts (1) and/or (2) except when b=-1 and d=-1.
其他解答:
i think this is the only answer, comparing terms....
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