Reduce this equation using the quadratic formula, factoring, or other algebraic method to determine the other zeros of the function.
F(x)= 4x^4 - x^3 - 11x^2 -2x +6
4x^4-11x^2+6=(x^2-2)(4x^2-3)
x^3-2x=x(x^2-2)
then you have a common factor, (x^2-2) that can be factored, and it will reduce rapidly
To find the zeros of the function F(x) = 4x^4 - x^3 - 11x^2 - 2x + 6, we need to solve the equation F(x) = 0. There are different methods we can use, such as factoring, the quadratic formula, or other algebraic methods.
Let's try factoring the equation first:
Step 1: Arrange the equation in descending order of powers of x:
4x^4 - x^3 - 11x^2 - 2x + 6 = 0
Step 2: Look for any common factors that we can factor out. In this case, there is no common factor that can be factored out.
Step 3: Try factoring by grouping or using trial and error. Unfortunately, the equation at hand does not lend itself easily to factoring by grouping or using simple trial and error.
Since factoring does not seem to be straightforward in this case, let's try to use another method, such as the quadratic formula.
Step 1: Identify the coefficients of the equation:
a = 4, b = -1, c = -11
Step 2: Apply the quadratic formula:
The quadratic formula states that the solutions to the quadratic equation ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying this formula to our equation, we have:
x = (-(-1) ± √((-1)^2 - 4 * 4 * (-11))) / (2 * 4)
= (1 ± √(1 + 176)) / 8
= (1 ± √177) / 8
So, we have found the first two zeros of the function.
To determine the other zeros of the function, we can use long division or synthetic division to reduce the equation further. Dividing the original polynomial by (x - k), where k is one of the two zeros we already found, will give us a quotient that is a lower-degree polynomial. By repeating this process, we can eventually reduce the polynomial to a quadratic or linear equation and find the remaining zeros.
However, since the quadratic formula already gave us the zeros of the equation, there are no other zeros to find in this case.
In summary, the zeros of the function F(x) = 4x^4 - x^3 - 11x^2 - 2x + 6 are given by the solutions of the quadratic equation (1 ± √177) / 8.