Step 1: Understanding the Question:
We need to identify which of the given linear binomials is a factor of the cubic polynomial $x^3 - 19x + 30$.
Step 2: Key Formula or Approach:
Use the Factor Theorem: A linear polynomial $(x - c)$ is a factor of a polynomial $P(x)$ if and only if $P(c) = 0$. We will substitute the root of each option into the polynomial and check which yields zero.
Step 3: Detailed Explanation:
Let the given polynomial be $P(x) = x^3 - 19x + 30$.
We test the root corresponding to each option by setting the option to zero ($x - c = 0 \Rightarrow x = c$).
Option (A) $x + 2$: Here $c = -2$.
$P(-2) = (-2)^3 - 19(-2) + 30 = -8 + 38 + 30 = 60 \neq 0$. Not a factor.
Option (B) $x + 1$: Here $c = -1$.
$P(-1) = (-1)^3 - 19(-1) + 30 = -1 + 19 + 30 = 48 \neq 0$. Not a factor.
Option (C) $x - 2$: Here $c = 2$.
$P(2) = (2)^3 - 19(2) + 30 = 8 - 38 + 30 = -30 + 30 = 0$.
Since $P(2) = 0$, by the Factor Theorem, $(x - 2)$ is indeed a factor of the polynomial.
We do not technically need to check further in a single-choice exam, but for completeness:
Option (D) $x - 1$: Here $c = 1$.
$P(1) = (1)^3 - 19(1) + 30 = 1 - 19 + 30 = 12 \neq 0$. Not a factor.
Option (E) $x - 7$: Here $c = 7$.
$P(7) = (7)^3 - 19(7) + 30 = 343 - 133 + 30 = 240 \neq 0$. Not a factor.
Step 4: Final Answer:
The factor is $x - 2$.