To find all the roots of the function f(x) = 5x^3 + 5x^2 – 170x + 280, given that one of its factors is (x – 7), we can use the Remainder Theorem. This theorem states that if a polynomial f(x) is divided by (x – c), the remainder of this division is f(c). Since x = 7 is a root, we know that f(7) = 0.
First, we will perform synthetic division to divide the polynomial by (x – 7).
7 | 5 5 -170 280
| 35 280
-------------------
5 40 110 0
The result of the division is 5x^2 + 40x + 110. Now, we need to find the roots of this quadratic equation.
We can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
For our equation, a = 5, b = 40, and c = 110.
Calculating the discriminant:
b² – 4ac = (40)² – 4(5)(110) = 1600 – 2200 = -600
Since the discriminant is negative, the quadratic has no real roots, but it does have two complex roots.
Calculating the roots using the quadratic formula:
x = (-40 ± √(-600)) / (2 * 5) = (-40 ± 10i√6) / 10 = -4 ± i√6
Therefore, the roots of the function f(x) = 5x^3 + 5x^2 – 170x + 280 are:
- x = 7
- x = -4 + i√6
- x = -4 – i√6