Final answer:
A quartic function with -2 and 1 as the only real zeros can be formulated by creating factors for these zeros and adding a pair of complex conjugates. The final function is derived by multiplying these factors together, resulting in a quartic polynomial.
Step-by-step explanation:
To find a quartic function with x-values -2 and 1 as its only real zeros, we can start by creating factors from these zeros. For the zero at x = -2, the corresponding factor is (x + 2), and for the zero at x = 1, the factor is (x - 1). Since the function is quartic, we need two more factors to ensure the degree of the function is four. These can be complex conjugates so they don't add any new real zeros.
Let's consider the complex pair (x + bi) and (x - bi), where 'b' is any real number. The product of these factors gives a quadratic with no real zeros: (x^2 + b^2). Multiplying all these factors together gives us:
f(x) = (x + 2)(x - 1)(x^2 + b^2)
We can expand this to get our final quartic function:
f(x) = x^4 + (2b^2 - 1)x^2 - 2x^3 - 2b^2x + 2x - 2b^2
To outline the steps used to find a quartic function:
- Create factors from the known real zeros.
- Introduce complex conjugate factors to add up to four factors.
- Multiply all factors to obtain the quartic function.