What are the elementary symmetric polynomials in roots r1,…,rn?
Solution
Correct Answer: Option A
Correct answer: Option 1.
Explanation:
The elementary symmetric polynomials in the roots \(r_1,\dots,r_n\) are defined for \(k=1,\dots,n\) 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.
Explanation:
The elementary symmetric polynomials in the roots \(r_1,\dots,r_n\) are defined for \(k=1,\dots,n\) by
\[
e_k(r_1,\dots,r_n)=\sum_{1\le i_1<\cdots
Thus
\[
e_1=\sum_{i=1}^n r_i,\qquad
e_2=\sum_{1\le i
e_n=r_1r_2\cdots r_n.
\]
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.
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