Question

What are the possible numbers of positive real, negative real, and complex zeros of
f(x) = -7x4 - 12x3 + 9x2 - 17x + 3?

Answer 1

Positive Real: 3 or 1
Negative Real: 1
Complex: 2 or 0

Positive Real: 3 or 1
Negative Real: 2 or 0
Complex: 1

Positive Real: 1
Negative Real: 3 or 1
Complex: 2 or 0

Positive Real: 4, 2 or 0
Negative Real: 1
Complex: 0 or 1 or 3

Answer 2

@ganeshie8 @KlOwNlOvE

Answer 3

Do you know Descartes' Rule of signs?

Answer 4

haven't really gone over it much yet

Answer 5

As the function is right now, use the rule and count how many sign changes there are, starting with the negative out front. How many sign changes are there?

Answer 6

3?

Answer 7

yes, there are 3. The rule says that you can have n number of changes and n - 2, n - 4, etc. Since there are 3 sign changes, you will have either 3 real roots or 1 real root. Let's make a table to chart out info.

Answer 8

This will be the chart, and now we will fill in the possible positive roots, or zeros. Do you know how to tell the number of roots possible for a function? What do you look at to see how many roots there are in general?

Answer 9

use the rational roots test?

Answer 10

No, the Rational Roots theorem tells you about what the possible factors of a polynomial are. It's much easier than that. Look at the highest power of the polynomial. What is it?

Answer 11

4

Answer 12

\[-7x ^{4} \]

Answer 13

?

Answer 14

sorry I lost the connection. I'm here! Yes, there are 4.

Answer 15

so that eliminates #3 and 4

Answer 16

right that eliminates #3 and #4.

Answer 17

now let's look at negative roots. Put in a -x for x and see if the sign changes. If the power is even the sign will not change; if it is odd, the sign will change.

Answer 18

-7x^4 - 12x^3 + 9x^2 - 17x + 3
so 1 power changes?