How many complex zeros does the polynomial function have? f(x)=−3x^6−2x^4+5x+6

Question

How many complex zeros does the polynomial function have?
f(x)=−3x^6−2x^4+5x+6

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    2021-09-09T16:16:20+00:00

    one way would be to factor

    I can’t factor it so we will have to use Descartes’ Rule of Signs which is helpful for finding how many real roots you have

    it goes like this:

    for a polynomial with real coefients, consider f(x)=-3x^6-2x^4+5x+6.

    after arranging the terms in decending order in terms of degree, count how many times the signs of the coeffients change direction and minus 2 from that number until you get to 1 or 0. that will be the number of even roots the function can have

    We have (-, -, +, +). the signs changed 1 times, so it has 1 real positive root

    to get the negative roots, we evaluate f(-x) and see how many times the root changes

    f(-x)=-3x^6-2x^4-5x+6

    signs are (-, -, -, +). there was 1 change in sign

    so the function has 1 real negative root

    a total of 2 real roots

    a function of degree n can have at most,  n roots

    our function is degree 6 so it has 6 roots

    if 2 are real, then the others must be complex

    6-2=4 so there are 4 complex roots

    you can also show that there are only 2 real roots by using a graphing utility to see that there are only 2 real roots

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