find all possible roots of the polynomial where x∈c:?

a) 2x^3+5x^2+14x+6=0

b) 8x^4=x

c) x^2(4x^2+17)=15

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  • 4 weeks ago
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    a)

    Let f(x) = 2x³ + 5x² + 14x + 6

    f(-1/2) = 0

    Then, (2x + 1) is a factor of 2x³ + 5x² + 14x + 6.

    2x³ + 5x² + 14x + 6 =0

    (2x³ + x²) + (4x² + 2x) + (12x + 6) = 0

    x²(2x + 1) + 2x(2x + 1) + 6(2x + 1) = 0

    (2x + 1)(x² + 2x + 6) = 0

    2x + 1 = 0  or  x² + 2x + 6 = 0

    x = -1/2

    (x² + 2x + 6 = 0 has no real roots because discriminant = -20 < 0)

    ====

    b)

    8x⁴ = x

    8x⁴ - x = 0

    x(8x³ - 1) = 0

    x[(2x)³ - 1³] = 0

    x(2x - 1)(4x² + 2x + 1) = 0

    x = 0  or  2x - 1 = 0  or  4x² + 2x + 1 = 0

    x = 0  or  x = 1/2

    (4x² + 2x + 1 = 0 has no real roots because discriminant = -12 < 0)

    ====

    c)

    x²(4x² + 17) = 15

    4x⁴ + 17x² - 15 = 0

    (4x² - 3)(x² + 5) = 0

    (2x + √3)(2x - √3)(x² + 5) = 0

    2x + √3 = 0  or  2x - √3 = 0  or  x² + 5 = 0

    x = -(√3)/2  or  x = (√3)/2

    (x² + 5 = 0 has no real roots.)

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