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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
Solution
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
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