Subject
গণিত
Topic
Algebra
What are the elementary symmetric polynomials in roots r1,…,rn?
What are the elementary symmetric polynomials in roots r1,…,rn?
ক)
e1 = Σri, e2 = Σ_{iখ)
e1 = Πri, e2 = Σri, …, en = Σ r_i^nগ)
e1 = Σ r_i^2, e2 = Σ r_i^3, …, en = Σ r_i^nঘ)
e1 = r1 + r2, e2 = r1r2 only for n = 2; undefined for n > 2Explanation
Correct answer: Option 1. Explanation: The elementary symmetric polynomials in the roots are defined for by \[ e_k(r_1,\dots,r_n)=\sum_{1\le i_1<\cdots \] Thus \[ e_1=\sum_{i=1}^n r_i,\qquad e_2=\sum_{1\le i \ldots,\qquad e_n=r_1r_2\cdots r_n. \] Examples: for \(n=3\), \[ e_1=r_1+r_2+r_3,\quad e_2=r_1r_2+r_1r_3+r_2r_3,\quad e_3=r_1r_2r_3. \] Why this is the standard choice: if \(p(x)\) is the monic polynomial with roots \(r_i\), \[ p(x)=\prod_{i=1}^n (x-r_i)=x^n-e_1x^{\,n-1}+e_2x^{\,n-2}-\cdots+(-1)^n e_n, \] so the coefficients of \(p\) are exactly the elementary symmetric polynomials (Vieta’s formulas). Why the other options are incorrect: - Option 2 mixes products and sums incorrectly (it places \(\prod r_i\) as \(e_1\) and uses power sums for later \(e_k\)); that does not match the definition above. - Option 3 gives power sums \(p_k=\sum_i r_i^k\), which are symmetric but are not the elementary symmetric polynomials. - Option 4 only describes the \(n=2\) case and does not generalize; elementary symmetric polynomials are defined for every \(n\) as in Option 1. Thus Option 1 is the correct description.Related questions
Define the arithmetic mean (AM) of positive numbers x1, …, xn.For ax2+ bx + c = 0 with roots r1, r2, state Vieta’s relations.For ax3+ bx2+ cx + d with roots r1, r2, r3, state Vieta’s relations.From x2− 5x + 6 = 0, find the roots using Vieta.Given roots 1 and −3 for a monic quadratic, find the polynomial.Real-coefficient polynomial with roots 2, 1 ± i; write the monic cubic.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.