l Descartes 4x^7+3x^6+x^5+2x^4-x^3+9x^2+x+1

Enter polynomial to factor:

    
Using Descartes' Rule of Signs
Find the number of real solutions for :
4x7 + 3x6 + x5 + 2x4 - x3 + 9x2 + x + 1

Evaluate the possible positive roots:
ƒ(x) = 4x7 + 3x6 + x5 + 2x4 - x3 + 9x2 + x + 1

There are 2 sign change(s):

Sign Change 1) + to -
Sign Change 2) - to +

Find more possible positive roots
Count down in pairs until we pass zero.
2 roots - 1 pair (2 roots) = 0


2 or 0 positive roots

Calculate possible negative roots:

Given ƒ(x) = 4x7 + 3x6 + x5 + 2x4 - x3 + 9x2 + x + 1
Determine ƒ(-x)
ƒ(-x) = 4(-x)7 + 3(-x)6 + (-x)5 + 2(-x)4 - (-x)3 + 9(-x)2 + (-x) + 1

-x raised to an even power is positive.
Odd exponents become negative:

4(-x)7
positive constant and odd exponent
We get a negative result of -4x7

3(-x)6
positive constant and even exponent
We get a positive result of + 3x6

(-x)5
positive constant and odd exponent
We get a negative result of - x5

2(-x)4
positive constant and even exponent
We get a positive result of + 2x4

-(-x)3
negative constant and odd exponent
We get a positive result of + x3

9(-x)2
positive constant and even exponent
We get a positive result of + 9x2

(-x)
positive constant and odd exponent
We get a negative result of - x

1
positive constant and even exponent
We get a positive result of + 1

ƒ(-x) = -4x7 + 3x6 - x5 + 2x4 + x3 + 9x2 - x + 1

Evaluate the possible negative roots:
ƒ(x) = - 4x7 + 3x6 - x5 + 2x4 + x3 + 9x2 - x + 1

There are 5 sign change(s):

Sign Change 1) - to +
Sign Change 2) + to -
Sign Change 3) - to +
Sign Change 4) + to -
Sign Change 5) - to +

Find more possible negative roots
Count down in pairs until we pass zero.
5 roots - 1 pair (2 roots) = 3
3 roots - 1 pair (2 roots) = 1


5 or 3 or 1 negative roots



(5 or 3 or 1) negative roots

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