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    Subject

    গণিত

    Topic

    Algebra

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

    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 r1,r2,r3r_1,r_2,r_3 of ax3+bx2+cx+d=0ax^3+bx^2+cx+d=0. By Viète's relations, r1+r2+r3=−ba,r1r2+r2r3+r3r1=ca,r1r2r3=−da. 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 r12+r22+r32=(r1+r2+r3)2−2(r1r2+r2r3+r3r1). 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: r12+r22+r32=(−ba)2−2(ca). 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: (−ba)2−2ca.\displaystyle\big(-\frac{b}{a}\big)^2-2\frac{c}{a}.

    Related questions

    What are the elementary symmetric polynomials in roots r1,…,rn?Express r1^2 + r2^2 for a quadratic ax^2 + bx + c = 0 in terms of coefficients.For monic x^3 + ax^2 + bx + c with roots r_i, give p1 = Σ r_i, p2 = Σ r_i^2, p3 = Σ r_i^3.For x2− Sx + P = 0 with nonzero roots r1, r2, find Σ 1/riand Σ 1/ri2.

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