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Polynomial equations

Rational roots, factoring by division and the fundamental theorem of algebra.

A polynomial of degree n has exactly n roots when complex ones and repeats are counted. Find rational roots by testing factors of the constant term over factors of the leading coefficient, divide them out, and finish with a quadratic.

Worked example: x^3 - 6x^2 + 11x - 6 = 0

Algebra · Polynomial-Equations

Solve x^3 - 6x^2 + 11x - 6 = 0

x^{3} - 6 x^{2} + 11 x - 6 = 0

Step by step

  1. x^{3} - 6 x^{2} + 11 x - 6 = 0

    Start from the equation as given.

  2. \left(x - 3\right) \left(x - 2\right) \left(x - 1\right) = 0

    Factor the polynomial (rational-root test + division).

  3. x - 1 = 0 \;\Rightarrow\; 1

    Set each factor equal to zero and solve it.

  4. x - 3 = 0 \;\Rightarrow\; 3

    Set each factor equal to zero and solve it.

  5. x - 2 = 0 \;\Rightarrow\; 2

    Set each factor equal to zero and solve it.

  6. x = 1 ,\; x = 3 ,\; x = 2

    All 3 roots.

Reveal the answer
x = 1 ,\; x = 3 ,\; x = 2

Try your own

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