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 + 1Determine 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)7positive 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
Common Core State Standards In This Lesson
HSA.SSE.A.2, HSA.APR.B.3
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.
* 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)
a3 + b3 = (a + b) (a2 - ab + b2)
a3 - b3 = (a - b) (a2 + ab + b2)
What 5 concepts are covered in the Factoring and Root Finding Calculator?
- factor
- a 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.
- polynomial
- an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
- rational expressions
- Quotient of two polynomials
P(x)/Q(x) - rational root theorem
- used to find the rational solutions of a polynomial equation
- rational zero theorem
- used to find the rational roots of a polynomial function
