AP EAMCET · Maths · Sequences and Series

Let $f (x) = x^{3} + a x^{2} + b x + c$ be polynomial with integer coefficients. If the roots of $f (x)$ are integer and are in Arithmetic Progression, then $' a'$ cannot take the value

A $- 642$
B $1214$
C $1323$
D $1626$

Verified Solution

Answer & Solution

Correct Answer

(B) $1214$

Step-by-step Solution

Detailed explanation

Given that, f(x)=x3+ax2+bx+c Now, since the roots of the polynomial are in Arithmetic Progression, Let the roots are s-t, s, s+t ⇒s-t+s+s+t=-a1 ⇒-3s=a Hence, a must be multiple of 3. From given options we see that only 1214 is not multiple of 3.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let f(x)=x3+ax2+bx+c be polynomial with integer coefficients. If the roots of f(x) are integer and are in Arithmetic Progression, then 'a' cannot take the value

Step-by-step Solution

Let $f (x) = x^{3} + a x^{2} + b x + c$ be polynomial with integer coefficients. If the roots of $f (x)$ are integer and are in Arithmetic Progression, then $' a'$ cannot take the value