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Given f(x) = 2x^3 + ax^2 - 7a^2x - 6a^3, determine whether or not x-a and x+a are factors of f(x). Hence, find, in terms of a, the roots of f(x) = 0.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "..........2x^2 + 3ax - 4a^2
\n" ); document.write( "x - a --> 2x^3 + ax^2 - 7(a^2)x - 6a^3
\n" ); document.write( "..........2x^3 - 2ax^2
\n" ); document.write( "...............3ax^2 - 7(a^2)x
\n" ); document.write( "...............3ax^2 - 3(a^2)x
\n" ); document.write( ".....................- 4(a^2)x - 6a^3
\n" ); document.write( ".....................- 4(a^2)x + 4a^3
\n" ); document.write( "...............................-10a^3 --> no on x - a
\n" ); document.write( "..........2x^2 - ax - 6a^2
\n" ); document.write( "x + a --> 2x^3 + ax^2 - 7(a^2)x - 6a^3
\n" ); document.write( "..........2x^3 + 2ax^2
\n" ); document.write( "...............-ax^2 - 7(a^2)x
\n" ); document.write( "...............-ax^2 - (a^2)x
\n" ); document.write( "....................- 6(a^2)x - 6a^3
\n" ); document.write( "....................- 6(a^2)x - 6a^3 --> yes on x + a
\n" ); document.write( "(x + a)(2x^2 - ax - 6a^2)
\n" ); document.write( "check: 2x^2(x + a) - ax(x + a) - 6a^2(x + a) =
\n" ); document.write( "2x^3 + 2ax^2 - ax^2 - (a^2)x - 6(a^2)x - 6a^3 =
\n" ); document.write( "2x^3 + ax^2 - 7(a^2)x - 6a^3
\n" ); document.write( "can 2x^2 - ax - 6a^2 be factored?
\n" ); document.write( "(x - 2a)(2x + 3a) --> check with FOIL --> 2x^2 + 3ax - 4ax - 6a^2 --> yes
\n" ); document.write( "f(x) = 2x^3 + ax^2 - 7(a^2)x - 6a^3 = 0
\n" ); document.write( "f(x) = (x + a)(x - 2a)(2x + 3a) = 0
\n" ); document.write( "x + a = 0 --> a = -x or x = -a
\n" ); document.write( "x - 2a = 0 --> -2a = -x --> 2a = x --> a = x/2 or x = 2a
\n" ); document.write( "2x + 3a = 0 --> 3a = -2x --> a = (-2/3)x or x = (-3/2)a
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