Find all the roots of f(x) = x^4-2x^3+5x^2-8x+4
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Rylan Hills
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The roots of f(x) = x^4-2x^3+5x^2-8x+4 have to be determined.
x^4-2x^3+5x^2-8x+4 = 0
=> x^4 - x^3 - x^3 + x^2 + 4x^2 - 4x - 4x + 4 = 0
=> x^3(x - 1) - x^2(x - 1) + 4x(x - 1) - 4(x - 1) = 0
=> (x^3 -x^2 + 4x - 4)(x - 1) = 0
=> (x^2(x - 1) + 4(x - 1))(x - 1) = 0
=> (x^2 + 4)(x - 1)^2 = 0
x - 1 = 0
=> x = 1
x^2 + 4 = 0
=> x = 2i and x = -2i
The roots of f(x) = x^4 - 2x^3 + 5x^2 - 8x + 4 are 2i, -2i and 1.
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