What are the possible number of positive, negative, and complex zeros of #f(x) = –3x^4 – 5x^3 – x^2 – 8x + 4#?
Look at changes of signs to find this has
Then do some sums...
#f(x) = 3x^45x^3x^28x+4#
Since there is one change of sign,
#f(x) = 3x^4+5x^3x^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+0.325x4.343#
#Delta = b^24ac = 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
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The possible number of positive, negative, and complex zeros of (f(x) = 3x^4  5x^3  x^2  8x + 4) can be determined using the rules of Descartes' Rule of Signs and the Fundamental Theorem of Algebra.

Descartes' Rule of Signs:
 The number of positive real zeros is either equal to the number of sign changes in the coefficients of (f(x)) or less than that by an even integer.
 The number of negative real zeros is either equal to the number of sign changes in (f(x)) or less than that by an even integer.
For (f(x)), there are 3 sign changes, so there are either 3 positive real zeros or 1 positive real zero (32=1). There are 2 sign changes in (f(x)), so there are either 2 negative real zeros or 0 negative real zeros (22=0).

Fundamental Theorem of Algebra:
 The number of complex zeros (including both real and imaginary parts) is equal to the degree of the polynomial, which is 4 in this case.
Combining the information from Descartes' Rule of Signs and the Fundamental Theorem of Algebra, we can conclude:
 There are 3 or 1 positive real zeros.
 There are 2 or 0 negative real zeros.
 There are 4 complex zeros.
Please note that these are the possible numbers of zeros, and the actual number of zeros may vary based on the specific roots of the polynomial.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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