Solving the Cubic Equation: $x^3 - 10x^2 + 31x - 30 = 0$

The goal is to identify which of the provided options are roots of the given cubic equation. A root of an equation is a value that, when substituted for the variable, makes the equation true.

Root Verification

We will test the options provided (1, 2, 3, 4) by substituting them into the equation $x^3 - 10x^2 + 31x - 30 = 0$.

• Test $x=1$: $(1)^3 - 10(1)^2 + 31(1) - 30 = 1 - 10 + 31 - 30 = -8$ Since $-8 \neq 0$, $x=1$ is not a root.
• Test $x=2$: $(2)^3 - 10(2)^2 + 31(2) - 30 = 8 - 10(4) + 62 - 30 = 8 - 40 + 62 - 30 = 70 - 70 = 0$ Since the result is $0$, $x=2$ is a root.
• Test $x=3$: $(3)^3 - 10(3)^2 + 31(3) - 30 = 27 - 10(9) + 93 - 30 = 27 - 90 + 93 - 30 = 120 - 120 = 0$ Since the result is $0$, $x=3$ is a root.
• Test $x=4$: $(4)^3 - 10(4)^2 + 31(4) - 30 = 64 - 10(16) + 124 - 30 = 64 - 160 + 124 - 30 = 188 - 190 = -2$ Since $-2 \neq 0$, $x=4$ is not a root.

Identifying Correct Roots

Based on the verification:

• $x=2$ satisfies the equation.
• $x=3$ satisfies the equation.

Therefore, the roots of the equation $x^3 - 10x^2 + 31x - 30 = 0$ from the given options are 2 and 3.