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8.4 SYMMETRIC FUNCTIONS
If we multiply out (x - a)(x - b)(x - c)(x - d), we obtain an expression of the form
x4 - s1x3 + s2x2 - s3x + s4, where
s1 = a + b + c + d, s2 = ab + ac + ad + bc + bd + cd, s3 = abc + abd + acd + bcd, s4 = abcd.We note that sk is the sum of all products of a, b, c, and d taken k at a time. It is also clear that the sk are symmetric functions of a, b, c, and d; that is, they do not change value when any two of a, b, c, and d are interchanged.
has a factorization into linear factors
where the ri are complex numbers. One can then see
that (-1)kak/a0 is the sum of all
the products of k factors chosen from r1, r2,
The absolute value of a real number x is written as |x| and is defined as follows: If then |x| = x; if x < 0, then |x| = -x.
Problems for Section 8.4
1. Let 3(x - r)(x - s) = 3x2 - 12x + 8. Find the following:
(a) r + s.
(c) (r + s)2.
(d) r2 + s2.
(e) r2 - 2rs + s2.
(f) |r - s|.
2. Find the sum, product, and absolute value of the difference of the roots of 5x2 + 7x - 4 = 0.
3. Let (x - r)(x - s) = x2 + x + 1. Show the following:
(a) r = -(s + 1), s = -(r + 1).
(b) r3 + r2 + r = 0 = s3 + s2 + s.
(c) r = s2, s = r2.
(d) r-1 + s2 = -1, s-1 + r2 = -1.
(e) r4 + r-1s-1 + s4 = 0.
(f) r9 - r6 + r3 - 1 = 0 = s9 - s6 + s3 - 1.
(g) r10 + s7 + r4 + s = -2 = s10 + r7 + s4 + r.
(h) (r2 - r + 1)(s2 - s
+ 1)(r4 - r2 + 1)(s4
- s2 + 1) = 16.
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