Answer:
Step-by-step explanation:
The factors of a polynomial function are the values that, when plugged into the function, result in an output of zero (i.e., make the function equal to zero).
To determine the factors using the rational root theorem, we examine the coefficients of the polynomial function and list all the possible rational roots by taking the factors of the constant term (the number at the end) divided by the factors of the leading coefficient (the coefficient of the highest power of x).
In this case, the constant term is -15 and the leading coefficient is 1, so we need to find the factors of -15 divided by the factors of 1. Here are all the possible rational roots:
±1, ±3, ±5, ±15
Now, we plug each of these values into the polynomial function and see if any of them make the function equal to zero.
f(1) = (1)^3 + 7(1)^2 + 7(1) - 15 = 1 + 7 + 7 - 15 = 0
f(-1) = (-1)^3 + 7(-1)^2 + 7(-1) - 15 = -1 + 7 - 7 - 15 = -16
f(3) = (3)^3 + 7(3)^2 + 7(3) - 15 = 27 + 63 + 21 - 15 = 96
f(-3) = (-3)^3 + 7(-3)^2 + 7(-3) - 15 = -27 + 63 - 21 - 15 = 0
f(5) = (5)^3 + 7(5)^2 + 7(5) - 15 = 125 + 175 + 35 - 15 = 320
f(-5) = (-5)^3 + 7(-5)^2 + 7(-5) - 15 = -125 + 175 - 35 - 15 = 0
f(15) = (15)^3 + 7(15)^2 + 7(15) - 15 = 3375 + 1575 + 105 = 5055
f(-15) = (-15)^3 + 7(-15)^2 + 7(-15) - 15 = -3375 + 1575 - 105 = -1905
From this analysis, we can see that two of the possible rational roots, x = 1 and x = -3, make the function equal to zero. Therefore, these are factors of the polynomial function f(x) = x^3 + 7x^2 + 7x - 15.
The factors can be written as:
(x - 1)(x + 3)