# 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