To calculate : The number of complex roots, the possible number of real roots and the possible rational roots.
The given polynomial has one complex root, three real roots, and no rational roots.
Given information :
Concept used:
Fundamental Theorem: If the degree of the polynomial is
Rational Root Theorem: For a given polynomial with integer coefficients, the rational roots will be in form of
Calculation :
To find the possible number of real roots use Descartes' rule.
The number of positive real zeroes is equal to the number of sign changes in consecutive coefficients of
So, there are two positive real zeros.
The number of negative real zeroes is equal to the number of sign changes in consecutive coefficients of
So, there is only one negative real zero.
Use the Fundamental Theorem to find the number of roots including both real and complex.
Here, the degree of polynomial is
Use Rational Root Theorem to find rational roots.
In the given polynomial, the constant term is
The factors of
So, the rational roots in term of
Use synthetic division method to test the roots.
As the remainder is not equal to zero, so
As the remainder is not equal to zero, so
As the remainder is not equal to zero, so
As the remainder is not equal to zero, so
Therefore, the given polynomial has one complex root, three real roots, and no rational roots.
Chapter 5 Solutions
High School Math 2015 Common Core Algebra 2 Student Edition Grades 10/11
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