To find the zeros of the polynomial function f(x) = 6x³ + 31x² + 4x – 5, we need to determine the values of x for which f(x) = 0.
The process of finding the zeros involves several steps:
- Identify the polynomial: The polynomial we are dealing with is a cubic polynomial.
- Use the Rational Root Theorem: This theorem suggests that potential rational roots of a polynomial can be found by taking the factors of the constant term and dividing them by the factors of the leading coefficient. In our case, the constant term is -5, and the leading coefficient is 6.
- List possible rational roots:
The factors of -5 are: ±1, ±5.
The factors of 6 are: ±1, ±2, ±3, ±6.
Hence, the possible rational roots could be:±1, ±5, ±1/2, ±5/2, ±1/3, ±5/3, ±1/6, ±5/6
- Test the potential roots:
Let’s substitute these values into the polynomial to see if any yield zero. After testing, you may discover that x = -1 is a root.
This can be verified by substituting -1 into the polynomial:
f(-1) = 6(-1)³ + 31(-1)² + 4(-1) – 5 = 0 - Polynomial Division: Once we find a root, we can perform synthetic or polynomial division to factor the polynomial. Using synthetic division with x + 1, we find:
f(x) = (x + 1)(6x² + 25x – 5) - Finding additional zeros: Next, we need to find the zeros of the quadratic component 6x² + 25x – 5 = 0. We can use the quadratic formula:
x = [-b ± sqrt(b² – 4ac)] / 2a, where a = 6, b = 25, c = -5. - Substituting the values into the formula gives us:
x = [-25 ± sqrt(25² – 4(6)(-5))] / (2 * 6)
x = [-25 ± sqrt(625 + 120)] / 12
x = [-25 ± sqrt(745)] / 12 - The zeros of the polynomial function, therefore, include:
x = -1, x = [-25 ± sqrt(745)] / 12
In summary, the zeros of the polynomial function f(x) = 6x³ + 31x² + 4x – 5 are:
- x = -1
- x = [-25 + sqrt(745)] / 12
- x = [-25 – sqrt(745)] / 12
These values represent the points where the polynomial intersects the x-axis.