To apply the rational root theorem, we need to consider all possible rational roots of the polynomial. The rational root theorem states that if a polynomial has a rational root p/q (where p and q are integers with no common factors), then p must be a factor of the constant term (25 in this case), and q must be a factor of the leading coefficient (3 in this case).
The possible rational roots of the polynomial f(x) = 3x^4– 26x^3 + 68x^2 –70x + 25 are determined by the factors of 25 (the constant term) divided by the factors of 3 (the leading coefficient). The factors of 25 are ±1, ±5, ±25, and the factors of 3 are ±1, ±3.
Therefore, the possible rational roots are:
±1/1, ±1/3, ±5/1, ±5/3, ±25/1, ±25/3
To find the rational roots, we can substitute each of these possible values into the polynomial and see if we get a zero result.
By testing these values, we find that the rational roots of the polynomial are:
x = 1 and x = 5/3
To find the complex roots, we can use polynomial division or synthetic division to divide the polynomial by (x - 1) and (x - 5/3) to obtain a quadratic equation.
Using synthetic division, we divide f(x) by (x - 1) to get:
3x^3 - 23x^2 + 45x - 25
Then, dividing the result by (x - 5/3), we get:
3x^2 - 20x + 15
Now, we can solve the quadratic equation 3x^2 - 20x + 15 = 0 to find the remaining complex roots.
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), where a = 3, b = -20, and c = 15, we have:
x = (20 ± √((-20)^2 - 4 * 3 * 15)) / (2 * 3)
x = (20 ± √(400 - 180)) / 6
x = (20 ± √220) / 6
x = (20 ± 2√55) / 6
x = (10 ± √55) / 3
Therefore, the complex roots of the polynomial are:
x = (10 + √55) / 3 and x = (10 - √55) / 3.
To summarize, the polynomial f(x) = 3x^4– 26x^3 + 68x^2 –70x + 25 has the following roots:
Rational roots: x = 1 and x = 5/3
Complex roots: x = (10 + √55) / 3 and x = (10 - √55) / 3.