Question

If the sum of two roots of the cubic equation $x^3 - 5x^2 - 2x + 24 = 0$ is $2$, then the roots of the equation are:

• A. $-2, 3, 4$
• B. $2, 3, 0$
• C. $-1, 3, 3$
• D. $-2, -3, -4$

Hint:

Use options to save time! Only option (A) contains the roots $-2, 3, 4$, which sum up to $5$ and have a pairwise product sum of $-2(3) + 3(4) + 4(-2) = -6 + 12 - 8 = -2$.

Answer 1

The Correct Option is A

Step 1: Use the root sum rule.
For a cubic $x^3 + bx^2 + cx + d = 0$, the sum of all three roots equals $-b$ (when the leading coefficient is $1$). For our equation $x^3 - 5x^2 - 2x + 24 = 0$, the sum of roots is $5$.

Step 2: Find the third root.
We are told two roots add to $2$. Since all three add to $5$, the third root is $5 - 2 = 3$.

Step 3: Use the product rule.
The product of all three roots equals $-d$, which is $-24$. So if the two unknown roots are $\alpha$ and $\beta$, then $\alpha\beta\times 3 = -24$.

Step 4: Find the product of the two roots.
Divide by $3$.
\[ \alpha\beta = \frac{-24}{3} = -8 \]

Step 5: Build a quadratic for them.
The two roots add to $2$ and multiply to $-8$. They satisfy $t^2 - 2t - 8 = 0$. Factorising gives $(t - 4)(t + 2) = 0$.

Step 6: List all roots.
So $t = 4$ or $t = -2$. Together with $3$, the roots are $-2, 3, 4$.
\[ \boxed{-2,\ 3,\ 4} \]