how many roots does this polynomial have: f(x) = 5x^3 + 8x^2 -4x + 3
and
Of the possible rational roots, which ones are roots: ±1, ±(1/5), ±3, ± (3/5)
According to the Fundamental Theorem of Algebra,
5x^3 + 8x^2 -4x + 3 =0 will have 3 roots, at least one of which must be real.
testing for x = 1
5 + 8 - 4 + 3 ≠ 0
testing for x = -1
-5 + 8 + 4 - 3 ≠ 0
tried the others, none worked, did not test the complex ones
Then tried Wolfram, go this
http://www.wolframalpha.com/input/?i=solve+5x%5E3+%2B+8x%5E2+-4x+%2B+3+%3D0
To determine the number of roots and to find the rational roots of a polynomial, we can use the Rational Root Theorem.
1. Number of Roots:
The degree of the polynomial is the highest power of x, which is 3 in this case. So, the polynomial f(x) = 5x^3 + 8x^2 - 4x + 3 is a cubic polynomial.
According to the Fundamental Theorem of Algebra, a cubic polynomial can have at most three roots. However, it doesn't necessarily mean that it will have three roots. It can also have fewer roots or even no real roots.
To find the number of real roots, we need to evaluate the polynomial at specific points. By checking the sign changes of the polynomial, we can determine the number of real roots.
For f(x) = 5x^3 + 8x^2 - 4x + 3:
- Substitute a large negative value such as x = -100 and calculate the value of f(x). If the value is positive, it counts as one sign. In this case, f(-100) = -5,079,997, so there is a sign change.
- Substitute a large positive value such as x = 100 and calculate f(x). If it is negative, it counts as another sign change. In this case, f(100) = 5,079,997, so there is a sign change.
By evaluating these two points, we find that there is exactly one sign change. Hence, there is exactly one real root.
2. Rational Roots:
To find the possible rational roots, we use the Rational Root Theorem. According to the theorem, if a rational number r is a root of the polynomial, then it must be of the form p/q, where p is a factor of the constant term (in this case, 3) and q is a factor of the leading coefficient (in this case, 5).
The possible rational roots for f(x) = 5x^3 + 8x^2 - 4x + 3 are calculated by considering all possible combinations of the factors of 3 and 5:
±1: Possible rational root
±(1/5): Possible rational root
±3: Possible rational root
±(3/5): Possible rational root
Now, by substituting these possible rational roots into the polynomial f(x) and checking if any of them result in zero, we can determine if they are actual roots.
By substituting these values, we find that f(1) = 12, f(-1) = -4, f(1/5) = 5/25 + 8/25 - 4/5 + 3 = 0, f(-1/5) = -5/25 + 8/25 + 4/5 + 3 = 0, f(3) = 492, f(-3) = -168, f(3/5) = 0, and f(-3/5) = 0.
From these evaluations, we find that ±1, ±(1/5), ±3, and ±(3/5) are the rational roots of the polynomial f(x) = 5x^3 + 8x^2 - 4x + 3.