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

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

What is the Answer?

(5 or 3 or 1) negative roots

How does the Factoring and Root Finding Calculator work?

Free Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:
* Difference of Squares
* Sum of Cubes
* Difference of Cubes
* Binomial Expansions
* Quadratics
* Factor by Grouping
* Common Term
This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots
* Factors and simplifies Rational Expressions of one fraction
* Determines the number of potential positive and negative roots using Descarte’s Rule of Signs
This calculator has 1 input.

What 3 formulas are used for the Factoring and Root Finding Calculator?

a2 - b2 = (a + b)(a - b)
a3 + b3 = (a + b) (a2 - ab + b2)
a3 - b3 = (a - b) (a2 + ab + b2)

For more math formulas, check out our Formula Dossier

What 5 concepts are covered in the Factoring and Root Finding Calculator?

factora divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n.polynomialan expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).rational expressionsQuotient of two polynomials
P(x)/Q(x)rational root theoremused to find the rational solutions of a polynomial equationrational zero theoremused to find the rational roots of a polynomial function

Example calculations for the Factoring and Root Finding Calculator

Factoring and Root Finding Calculator Video

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