What is a quartic function with only the two real zeros given?
x = -4, x = 1
To determine a quartic function with the given real zeros, we need to determine the equations for two quadratic factors and multiply them together.
First, let's find the equations for the quadratic factors:
For the real zero x = -4, the factor is (x + 4).
For the real zero x = 1, the factor is (x - 1).
Multiplying these two factors together, we get:
F(x) = (x + 4)(x - 1)
Expanding the equation:
F(x) = x^2 - x + 4x - 4
Simplifying the equation:
F(x) = x^2 + 3x - 4
Therefore, a quartic function with the two real zeros x = -4 and x = 1 is F(x) = x^2 + 3x - 4.
To find a quartic function with the given real zeros, we can use the fact that zeros of a function correspond to its factors.
Since x = -4 and x = 1 are both zeros of the quartic function, the factors can be expressed as (x - (-4)) and (x - 1). Simplifying this, we have (x + 4) and (x - 1).
For a quartic function, we need to have four factors. To find the other two factors, we can choose any two additional zeros. Let's say we choose x = 2 and x = -3.
Now, our four factors would be (x + 4), (x - 1), (x - 2), and (x + 3).
To find the quartic function, we multiply these factors together:
f(x) = (x + 4)(x - 1)(x - 2)(x + 3)
Expanding this expression gives the quartic function with the given real zeros:
f(x) = (x^2 + 3x - 4)(x^2 + x - 6)
(x+4)(x-1) = 0
y = x^2 + 3 x - 4