Algebra 2 Descartes' Rule of Signs

[Loonygirl]

Use Descartes' Rule of Signs to analyze the zeros of the following. List all possibilities.

1. P(x) = x5 - 4x4 + 3x3 + 2x - 6

2. P(x) = -5x4 + x3 + 2x2 - 1

3. P(x) = 2x5 + 7x3 + 6x2 - 2





So I found these answers, are they right?:

1. P(x) = x^5 - 4x^4 + 3x^3 + 2x - 6
(3 positive, 2 imaginary) ; (1 positive, 4 imaginary)

2. P(x) = -5x^4 + x^3 + 2x^2 - 1
(2 positive, 2imaginary) ; (4imaginary) ; (2 positive, 2 negative) ; (2 neg. 2 imagin.)

3. P(x) = 2x^5 + 7x^3 + 6x^2 - 2
(1 positive, 4 imaginary) ; (1 positive, 2 negative, 2 imaginary)

 

[HallsofIvy]

All DesCarte's rule of signs says is "if the number of changes from signs of coefficients, from leading term down is n, they there are n, n-2, n-4, ... down to 0 positive roots. If you change the sign on all odd powers, you swap positive and negative roots so now the same is true for negative roots."

For 1, P(x)= x^5- 4x^4+ 3x^3+ 2x- 6, there are 3 changes of sign while P(x)= -x^5- 4x^3- 3x^3- 2x- 6 there is no change of sign. That means there are no negative roots and there can be either 3 or 1 positive roots. If there are 3 positive roots but no negative roots, the other 2 roots must be non-real. If there is only one root but no negative roots, tghe other 4 roots must be non-real. You are exactly right except that I would not use the word "imaginary" here. To me that means a number of the form bi rather than a+ bi. I started to say "complex" but that includes the real numbers!

The others look good also.

 

[epenguin]

I would not use the word "imaginary" here. To me that means a number of the form bi rather than a+ bi. I started to say "complex" but that includes the real numbers!

I and everybody (?) used to say 'complex' but at some point i picked up by ear you are supposed to say 'nonreal' for complex numbers a+bi where b not = 0.

Am I right and can anyone tell me tex for 'not equal'?

 

[HallsofIvy]

I and everybody (?) used to say 'complex' but at some point i picked up by ear you are supposed to say 'nonreal' for complex numbers a+bi where b not = 0.

Am I right and can anyone tell me tex for 'not equal'?

The complex numbers include the real numbers. "4" is just as much a complex number as "4i" or "3+ 4i" are.

And the tex for "not equal" is "\ne": $\ne$.