What are the possible numbers of positive, negative, and complex zeros of f(x) = 3x4 − 5x3 − x2 − 8x + 4?

Positive: 2; negative: 2; complex: 4 or 2 or 0

Positive: 1; negative: 3 or 1; complex: 2 or 0

Positive: 3 or 1; negative: 1; complex: 2 or 0

Positive: 2 or 0; negative: 2 or 0; complex: 4 or 2 or 0

yes, descartes rule of signs

for positive roots
see how many times it changes
+3x^4-5x^3-x^2-8x+4
+,-,-,-,+
2 changes
2 or 0 posotive roots

for negative
sub every x with -x
3x^4+5x^3-x^2+8x+4
+,+,-,+,+
2 changes
2 or 0 negative roots

not sure how to find complex roots

answer is last one

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Blitztiger Blitztiger · Answer 2 · 2018-07-11T04:04:32-04:00

Using Descartes' Rule of Signs, we observe whether each consecutive term has a + or - sign:
f(x) = 3x4 − 5x3 − x2 − 8x + 4

Signs are: + - - - +

There are 2 sign changes (from +3 to -5, then from -8 to +4). This means that there are either 2 or 0 positive roots. (We take the number of sign changes, which is 2, and subtract by 2s to get all possible numbers of positive roots.)

For negative roots, we first invert all the odd-powered terms in f(x):

3x4 + 5x3 − x2 + 8x + 4

Signs are: + + - + +
There are 2 sign changes, from +5 to -1, and from -1 to +8. Therefore, there are either 2 or 0 negative roots.
Since the overall degree of f(x) is 4, the total number of roots must sum up to 4. There are either 2 or 0 positive roots, and 2 or 0 negative roots, and everything that remains will be a complex root. The possibilities are:
2 positive, 2 negative, 0 complex
2 positive, 0 negative, 2 complex
0 positive, 2 negative, 2 complex
0 positive, 0 negative, 4 complex.
Therefore we can have either 0, 2, or 4 complex roots.

The correct answer is the last choice.