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Question 389398: 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.
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! 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.
..........2x^2 + 3ax - 4a^2
x - a --> 2x^3 + ax^2 - 7(a^2)x - 6a^3
..........2x^3 - 2ax^2
...............3ax^2 - 7(a^2)x
...............3ax^2 - 3(a^2)x
.....................- 4(a^2)x - 6a^3
.....................- 4(a^2)x + 4a^3
...............................-10a^3 --> no on x - a
..........2x^2 - ax - 6a^2
x + a --> 2x^3 + ax^2 - 7(a^2)x - 6a^3
..........2x^3 + 2ax^2
...............-ax^2 - 7(a^2)x
...............-ax^2 - (a^2)x
....................- 6(a^2)x - 6a^3
....................- 6(a^2)x - 6a^3 --> yes on x + a
(x + a)(2x^2 - ax - 6a^2)
check: 2x^2(x + a) - ax(x + a) - 6a^2(x + a) =
2x^3 + 2ax^2 - ax^2 - (a^2)x - 6(a^2)x - 6a^3 =
2x^3 + ax^2 - 7(a^2)x - 6a^3
can 2x^2 - ax - 6a^2 be factored?
(x - 2a)(2x + 3a) --> check with FOIL --> 2x^2 + 3ax - 4ax - 6a^2 --> yes
f(x) = 2x^3 + ax^2 - 7(a^2)x - 6a^3 = 0
f(x) = (x + a)(x - 2a)(2x + 3a) = 0
x + a = 0 --> a = -x or x = -a
x - 2a = 0 --> -2a = -x --> 2a = x --> a = x/2 or x = 2a
2x + 3a = 0 --> 3a = -2x --> a = (-2/3)x or x = (-3/2)a
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