Question

Show that `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|` is divisiible by x⁴ 

Answer 1

Let Δ = `|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|`

Δ = `"a"/"a"|("a"^2 + x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|`

Multiply C₁ by a

Δ = `1/"a"|("a"^2 + "a"x^2, "ab", "ac"),("ab", "b"^2 + x^2, "bc"),("ac", "bc", "c"^2 + x^2)|`

Applying `"C"_1 -> "C"_1 + "bC"_2 + "cC"_3`

= `1/"a"|("a"^3 + "a"x^2 +"ab"^2 + "ac"^2, "ab", "ac"),("a"^2"b" + "b"^3 + "b"x^2 + "bc"2, "b"^2 + x^2, "bc"),("a"^2"c" + "b"^2"C" + "c"^3 + "c"x^2, "bc", "c"^2 + x^2)|`

= `1/"a"|("a"("a"^2 + "b"^2 + "c"^2 + x^2), "ab", "ac"),("b"("a"^2 + "b"^2 + "c"^2 + x^2), "b"^2 + x^2, "bc"),("c"("a"^2 + "b"^2 + "c"^2 + x^), "bc", "c"^2 + x^2)|`

= `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" |("a", "bc", "ac"),("b","b"^2 + x^2, "bc"),("c", "bc", "c"^2 + x^2)|`

Applyig `"C"_2 -> "C"_2 - "bC"_1` and `"C"_3 -> "C"_3- "cC"_1`

= `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" |("a", "ab" - "ab", "ac" - "ac"),("b", "b"^2 + x^2 - "b"^2, "bc" - "bc"),("c", "bc" - "bc", "c"^2 + x^2 - "c"^2)|`

= `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" |("a", 0, 0),("b", x^2, 0),("c", 0, x^2)|`

Expanding along the first row

=  `("a"^2 + "b"^2 + "c"^2 + x^2)/"a" xx "a"[(x^2) (x^2) - (0) (0)] + 0 + 0`

= `("a"^2 + "b"^2+ "c"^2 + x^2)x^4`

Which is divisible by x⁴