For monic x^3 + ax^2 + bx + c with roots… | Answer

Subject

গণিত

Topic

Algebra

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.
Reading

ক)

p1 = −a; p2 = a^2 − 2b; p3 = −a^3 + 3ab − 3c

খ)

p1 = a; p2 = a^2 + 2b; p3 = a^3 + 3ab + 3c

গ)

p1 = −a; p2 = a^2 + 2b; p3 = −a^3 − 3ab + 3c

ঘ)

p1 = −b; p2 = b^2 − 2c; p3 = −b^3 + 3bc − 3a

Explanation

Let the monic cubic be

f(x)=x^{3}+ax^{2}+bx+c

with roots

r_{1},r_{2},r_{3}

. By Viète's relations

e_{1}=r_{1}+r_{2}+r_{3}=-a,\qquad e_{2}=r_{1}r_{2}+r_{2}r_{3}+r_{3}r_{1}=b,\qquad e_{3}=r_{1}r_{2}r_{3}=-c.

Denote

p_{k}=r_{1}^{k}+r_{2}^{k}+r_{3}^{k}

. Then 1)

p_{1}=e_{1}=-a.

2) For

p_{2}

use the square of the sum: \[ p_{2}=(r_{1}+r_{2}+r_{3})^{2}-2\sum_{i \] Substituting \(e_{1}=-a,\ e_{2}=b\) gives \[ p_{2}=(-a)^{2}-2b=a^{2}-2b. \] 3) For \(p_{3}\) use Newton's identity for degree 3: \[ p_{3}=e_{1}p_{2}-e_{2}p_{1}+3e_{3}. \] Substitute the known values: \[ p_{3}=(-a)(a^{2}-2b)-b(-a)+3(-c)=-a^{3}+2ab+ab-3c=-a^{3}+3ab-3c. \] Thus \[ p_{1}=-a,\qquad p_{2}=a^{2}-2b,\qquad p_{3}=-a^{3}+3ab-3c, \] which matches Option 1.

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 a cubic ax^3 + bx^2 + cx + d = 0, express Σ r_i^2 via coefficients.For x2− Sx + P = 0 with nonzero roots r1, r2, find Σ 1/riand Σ 1/ri2.

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.Reading

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.

Explanation

Related questions

Find Tutors

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.
Reading