Question

The equation \(x^{3}+5x^{2}+px+q=0\) and \(x^{3}+7x^{2}+px+r=0\) have two roots in common. If the third root of each equation is represented by \(x_{1}\) and \(x_{2}\) respectively, then \(\gcd(x_{1},x_{2})\) will be:

• A. 3
• B. 1
• C. $p$
• D. 2

Hint:

Whenever two polynomial equations share the exact same leading terms and linear terms (like $x^3$ and $px$ here), subtracting the two equations is an excellent way to simplify the problem. Subtracting them here gives $2x^2 + (r-q) = 0$, which immediately tells you that the two shared roots must be symmetric around zero ($\alpha = -\beta \implies \alpha+\beta=0$)!

Answer 1

The Correct Option is B

Concept: According to Vieta's formulas for a cubic polynomial equation $Ax^3 + Bx^2 + Cx + D = 0$, the sum of all three roots is equal to $-\frac{B}{A}$. By tracking the sum of the roots for two equations that share two identical roots, we can isolate the individual values of their remaining unique third roots. Step 1: Define the shared roots and apply Vieta's sum property.
Let the two common real roots shared by both cubic equations be defined as $\alpha$ and $\beta$.
• For the first cubic equation $x^{3}+5x^{2}+px+q=0$, the roots are $\alpha, \beta,$ and $x_1$. Applying the root-sum identity: \[ \alpha + \beta + x_1 = -\frac{5}{1} = -5 \quad \cdots (1) \]
• For the second cubic equation $x^{3}+7x^{2}+px+r=0$, the roots are $\alpha, \beta,$ and $x_2$. Applying the root-sum identity: \[ \alpha + \beta + x_2 = -\frac{7}{1} = -7 \quad \cdots (2) \]

Step 2: Isolate the shared components using subtraction.
We can eliminate the unknown shared root sum $(\alpha + \beta)$ by subtracting equation (1) from equation (2): \[ (\alpha + \beta + x_2) - (\alpha + \beta + x_1) = -7 - (-5) \] \[ x_2 - x_1 = -2 \quad \Rightarrow \quad x_1 - x_2 = 2 \quad \cdots (3) \]

Step 3: Analyze the shared linear coefficient condition.
Now let us look at the coefficient of the linear $x$ term, which is $p$ in both equations. According to Vieta's rule for the sum of the products of roots taken two at a time ($\sum \alpha\beta = \frac{C}{A}$):
• From the first equation: $\alpha\beta + x_1(\alpha + \beta) = p$
• From the second equation: $\alpha\beta + x_2(\alpha + \beta) = p$ Equating both expressions since they both equal $p$: \[ \alpha\beta + x_1(\alpha + \beta) = \alpha\beta + x_2(\alpha + \beta) \quad \Rightarrow \quad x_1(\alpha + \beta) = x_2(\alpha + \beta) \] \[ (x_1 - x_2)(\alpha + \beta) = 0 \]

Step 4: Calculate the explicit third roots and determine their GCD.
From equation (3), we know that $x_1 - x_2 = 2 \neq 0$. Therefore, for the product to equal zero, the sum component must vanish: \[ \alpha + \beta = 0 \] Substitute $\alpha + \beta = 0$ back into our primary root-sum equations (1) and (2): \[ 0 + x_1 = -5 \quad \Rightarrow \quad x_1 = -5 \] \[ 0 + x_2 = -7 \quad \Rightarrow \quad x_2 = -7 \] The unique third roots are exactly $x_1 = -5$ and $x_2 = -7$. Since $-5$ and $-7$ are distinct prime integers, they share no common numerical factors other than 1. Thus, their Greatest Common Divisor (GCD) is: \[ \text{GCD}(-5, -7) = 1 \] This matches option (B) perfectly.