A Polynomial Equation | Solving Special Cubics đ
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- Äas pĆidĂĄn 20. 06. 2024
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Fun I've been practicing ones like this. I noted that if x is real it's got to be negative, and hopefully it's a factor of 4. Once I found that (x + 2) is a factor of the cubic, it was easy there on.
Plz upload videos on linear algebra, vector , and matrix
you could also use the rat'l root thm and synthetic division. It takes 4 lines.
I wonder why high school math teachers never taught his class rational root theorem of factors of the last coefficient "plug and try" to know x= -2 is a factor. Then - 2 Synthetic divides into 1 1 0 4 as x^2 - x + 2 and then 0 remainder. So by high school Junior or Sophomore coursework students quickly obtain (x + 2)(x^2 - x + 2) that can be easily checked by multiplying those two factors with the last factor a tricky quadratic factor.
3rd method (incomplete): Observe that x cannot be zero. Divide through by 4x^3. Substitute y = 1/x and now the cubic in y is in the correct form for Cardano's method.
Due to small constant, by scrutinizing the equation it is clear that x=-2
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xÂł+xÂČ+4=(xÂČ+ax+b)(x+c)
=xÂł+(c+a)xÂČ+(ac+b)x+bc
c+a=1 (1)
ac+b=0 (2)
bc=4 (3)
Multiply (2) by c: acÂČ+bc=0
Noting (3) acÂČ+4=0
ac=-4 (4)
Multiply (1) by cÂČ: cÂł+acÂČ=cÂČ
Noting (4) cÂł-4=cÂČ
cÂČ(c-1)=4
=2ÂČ(2-1)
c=2, b=2, a=-1
Thus 0=xÂł+xÂČ+4
=(xÂČ--x+2)(x+2)
x=-2 or xÂČ-x+2=0
x=œ[1±sqrt(-7)]
=œ[1±isqrt(7)]
Nice