Algebra 2 Descartes' Rule of Signs

In summary, Descartes' Rule of Signs is a mathematical rule used in Algebra 2 to determine the possible number of positive and negative roots of a polynomial equation without actually solving it. This rule is used by counting the number of sign changes in the equation and comparing it to the number of positive and negative roots. It is important because it saves time and effort when dealing with complex equations. While it can be used for all polynomial equations in standard form, it can only determine the maximum number of roots and not their exact values.
  • #1
Loonygirl
5
0
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)
 
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  • #2
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.
 
  • #3
HallsofIvy said:
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'?
 
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  • #4
epenguin said:
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": [itex]\ne[/itex].
 

Related to Algebra 2 Descartes' Rule of Signs

What is Descartes' Rule of Signs in Algebra 2?

Descartes' Rule of Signs is a mathematical rule used to determine the number of possible positive and negative roots of a polynomial equation. It helps to identify the maximum number of positive and negative roots of a polynomial equation without actually finding the roots.

How is Descartes' Rule of Signs used in Algebra 2?

Descartes' Rule of Signs is used by counting the number of sign changes in a polynomial equation and then comparing it to the number of positive and negative roots. This rule can help to narrow down the possible number of roots without having to solve the equation.

What is the importance of Descartes' Rule of Signs in Algebra 2?

Descartes' Rule of Signs is important because it allows us to determine the possible number of roots of a polynomial equation without actually solving the equation. This can save time and effort when dealing with complex equations.

Can Descartes' Rule of Signs be used for all polynomial equations in Algebra 2?

Yes, Descartes' Rule of Signs can be used for all polynomial equations, including those with multiple variables and exponents, as long as the equation is in standard form.

Can Descartes' Rule of Signs determine the exact number of roots in a polynomial equation?

No, Descartes' Rule of Signs can only determine the maximum number of positive and negative roots in a polynomial equation. It cannot determine the exact number of roots or their values.

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