Cubic function

In mathematics, a cubic function is a function of the form

Root-finding formula

The formula for finding the roots of a cubic function is fairly complicated. Therefore, it is common for some students to use the rational root test or a numerical solution instead.

Related Topics:
Rational root test - Numerical solution

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If we have

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:f(x) = ax^3 + bx^2 + cx + d = a(x - x_1)(x - x_2)(x - x_3),

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Let

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:q = rac{{3c-b^2}}{{9}} and

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:r = rac{{9bc - 27d - 2b^3}}{{54}}

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Now, let

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:s = sqrt{{rac{{r}}{{2}} + sqrt{{rac{{q^3}}{{27}}+rac{{r^2}}{{4}}}}}} and

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:t = sqrt{{rac{{r}}{{2}} - sqrt{{rac{{q^3}}{{27}}+rac{{r^2}}{{4}}}}}}

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The solutions are

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:x_1 = s+t-rac{{b}}{{3}}

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:x_2=-rac{{1}}{{2}}(s+t)-rac{{b}}{{3}}+rac{{sqrt{{3}}}}{{2}}(s-t)i

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:x_3=-rac{{1}}{{2}}(s+t)-rac{{b}}{{3}}-rac{{sqrt{{3}}}}{{2}}(s-t)i

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See also: cubic equation, spline.

Related Topics:
Cubic equation - Spline

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