For a cubic ax^3 + bx^2 + cx + d = 0, exp… | Answer | TuitionWave

Subject

গণিত

Topic

Algebra

For a cubic ax^3 + bx^2 + cx + d = 0, express Σ r_i^2 via coefficients.
Reading

ক)

(−b/a)^2 − 2(c/a)

খ)

(−b/a)^2 + 2(c/a)

গ)

−b/a − 2c/a

ঘ)

(c/a)^2 − 2(b/a)

Explanation

Let the roots be

r_1,r_2,r_3

of

ax^3+bx^2+cx+d=0

. By Viète's relations,

r_1+r_2+r_3=-\frac{b}{a},\qquad r_1r_2+r_2r_3+r_3r_1=\frac{c}{a},\qquad r_1r_2r_3=-\frac{d}{a}.

Use the identity

r_1^2+r_2^2+r_3^2=(r_1+r_2+r_3)^2-2(r_1r_2+r_2r_3+r_3r_1).

Substitute the Viète expressions:

r_1^2+r_2^2+r_3^2=\Big(-\frac{b}{a}\Big)^2-2\Big(\frac{c}{a}\Big).

Thus the correct choice is Option 1:

\displaystyle\big(-\frac{b}{a}\big)^2-2\frac{c}{a}.

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