Question

Which polynomial function has a leading coefficient of 1 and roots (7+) and (5-) with multiplicity 1? 1) f(x) = (x + 7)(x-1)(x + 5)(x + 1) 2) f(x) = (x-7)(x-7)(x-5)(x + 1) 3) f(x) = (x-7 - √7)(x - √5)(x - 1)(x + √7)(x + √5) 4) f(x) = (x + 7 - √7)(x + √5)(x + 1)(x + √7)(x - √5)

Answer 1

The correct polynomial function is 1) f(x) = (x + 7)(x - 1)(x + 5)(x + 1).

Step-by-step explanation:

The leading coefficient of a polynomial is the coefficient of the term with the highest power of the variable. In this case, the leading coefficient is 1, indicating that the term with the highest power is simply x to the power of some non-negative integer.

The given roots are (7+) and (5-) with multiplicity 1. The notation (7+) implies a root at x = 7, and (5-) implies a root at x = 5. Multiplicity 1 means that each root occurs only once.

Examining option 1), f(x) = (x + 7)(x - 1)(x + 5)(x + 1), we find that it satisfies all the given conditions. The leading coefficient is 1, and it has roots at x = 7, x = 5, x = 1, and x = -1 with multiplicity 1.

Options 2), 3), and 4) do not have a leading coefficient of 1, and they do not have the correct combination of roots and multiplicities specified in the question. Therefore, option 1) is the correct polynomial function.