What are the possible number of positive, negative, and complex zeros of #f(x) = –3x^4 – 5x^3 – x^2 – 8x + 4#?
1 Answer
Look at changes of signs to find this has
Then do some sums...
Explanation:
#f(x) = -3x^4-5x^3-x^2-8x+4#
Since there is one change of sign,
#f(-x) = -3x^4+5x^3-x^2+8x+4#
Since there are three changes of sign
Since
Newton's method can be used to find approximate solutions.
Pick an initial approximation
Iterate using the formula:
#a_(i+1) = a_i - f(a_i)/(f'(a_i))#
Putting this into a spreadsheet and starting with
#x ~~ 0.41998457522194#
#x ~~ -2.19460208831628#
We can then divide
Notice the remainder
Check the discriminant of the approximate quotient polynomial:
#-3x^2+0.325x-4.343#
#Delta = b^2-4ac = 0.325^2-(4*-3*-4.343) = 0.105625 - 52.116 = -52.010375#
Since this is negative, this quadratic has no Real zeros and we can be confident that our original quartic has exactly