State the number of complex roots,possible number of real roots and possible rational roots for x^5 - x^3 - 11x^2+ 9x+18=0
there are 5 possible rational and real roots. Given the wording of the problem, though, it's likely that there are some complex roots, reducing the number of real candidates.
There may be 2 or 4 complex roots, meaning that there may be 5,3, or 1 real root.
Lacking any further information, all the real roots may be rational, and in fact will be integers.
A little synthetic division shows that we have no rational roots.
In fact, there are no real roots greater than 0 or less than -1. So, there are 4 complex roots and one real root -1 < r < 0
To determine the number of complex roots, possible number of real roots, and possible rational roots for the equation x^5 - x^3 - 11x^2 + 9x + 18 = 0, we can use a combination of techniques from algebra and calculus.
1. Complex roots:
The Fundamental Theorem of Algebra states that a polynomial with degree "n" has exactly "n" complex roots (counted with multiplicity). In this case, the degree of the polynomial is 5, so it will have exactly 5 complex roots.
2. Real roots:
To determine the number of real roots, we can examine the behavior of the polynomial function using calculus. However, it is not always easy to find the exact roots using calculus.
A simpler method is to apply Descartes' Rule of Signs. Count the number of sign changes in the coefficients of the polynomial when you write it in standard form:
From x^5 to -x^3, there is a sign change (positive to negative).
From -x^3 to -11x^2, there is no sign change.
From -11x^2 to 9x, there is a sign change (negative to positive).
From 9x to 18, there is no sign change.
So, according to the rule, there are either 2 or 0 positive real roots. Since the polynomial has degree 5, we know it must have either 3 or 1 negative real roots. Therefore, the possible number of real roots can be 0, 1, 2, or 3.
3. Rational roots:
To find the possible rational roots, we can use the Rational Root Theorem. This theorem states that any rational root of the form p/q, where p is a factor of the constant term (18) and q is a factor of the leading coefficient (1), will satisfy the equation if it exists.
The factors of 18 are: ±1, ±2, ±3, ±6, ±9, ±18.
The factors of 1 are: ±1.
By trying all these possible combinations (with both positive and negative signs), we can find any rational roots that satisfy the equation. In this case, it is necessary to use either a graphing calculator or software, or an advanced computational method like Newton's method to find the actual solutions.
In summary:
- The polynomial has 5 complex roots.
- The possible number of real roots is 0, 1, 2, or 3.
- The possible rational roots can be calculated by using the Rational Root Theorem, testing the factors of the constant term divided by the factors of the leading coefficient.