Question

Given the degree of 4 with zeros -5, 4, and -2 (multiplicity 2) and with end behavior as x approaches positive infinity, y approaches negative infinity, and as x approaches negative infinity, y approaches negative infinity, write an equation in factored form, and state if the function is even, odd, or neither even nor odd.

Answer 1

The polynomial equation is written in factored form as y = -(x + 5)(x - 4)(x + 2)^2, and given the end behavior, it is neither an even nor an odd function.

Step-by-step explanation:

The equation of the polynomial with a degree of 4 and zeros at -5, 4, and -2 (with a multiplicity of 2) can be written as:

y = a(x + 5)(x - 4)(x + 2)^2

Given the end behavior as x approaches positive infinity, y approaches negative infinity, and as x approaches negative infinity, y also approaches negative infinity, we know the leading coefficient a must be negative. To satisfy this condition, the coefficient a can be any negative number; for simplicity, we can choose a = -1:

y = -(x + 5)(x - 4)(x + 2)^2

Regarding the function's parity, the function is neither even nor odd since it does not fulfill the symmetry requirements for either even or odd functions. As the rules state, an even function is symmetric about the y-axis, which this function is not; an odd function exhibits origin symmetry, which is also not the case for this function.