To find the rational zeros of the polynomial function f(x) = x^3 - 43x + 42, we can use the Rational Root Theorem, which states that if a rational number p/q (where p and q are integers) is a zero of the polynomial, then p must be a factor of the constant term (42 in this case), and q must be a factor of the leading coefficient (1 in this case).
(a) Rational zeros of f(x) = x^3 - 43x + 42:
The factors of 42 are ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42.
The factors of 1 (leading coefficient) are ±1.
Possible rational zeros:
±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42.
To find the actual rational zeros, we can try each of these values by substituting them into f(x) and checking if the result is zero.
By trying out these values, we find that the rational zeros of f(x) are:
x = -1, x = 1, x = -2, x = 3, x = 6, and x = 7.
(b) Factoring f(x) into linear factors:
Now that we have found the rational zeros of f(x), we can factor f(x) into linear factors using these zeros. The linear factors can be written as (x - zero):
f(x) = (x - (-1))(x - 1)(x - (-2))(x - 3)(x - 6)(x - 7)
Simplifying this expression, we get:
f(x) = (x + 1)(x - 1)(x + 2)(x - 3)(x - 6)(x - 7)
So, f(x) is factored into linear factors as (x + 1)(x - 1)(x + 2)(x - 3)(x - 6)(x - 7).
I hope this helps!!