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Solve x³ - 6x² + 5x = 0

The solutions of x³ - 6x² + 5x = 0 are x = 0, 1, 5.

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Graph of y = x³ - 6x² + 5x-1123456-20-101020304050x = 0x = 1x = 5y
Graph of y = x³ - 6x² + 5x. The curve crosses the x-axis at x = 0, x = 1, x = 5, the roots found above.

How to work it out

  1. 1

    A cubic has up to 3 real roots. The rational root theorem says that any integer root of a polynomial with integer coefficients and leading coefficient 1 divides the constant term 0 (it is 0, so x = 0 is a root and x factors out).

    x3−6x2+5x=0x^{3} - 6x^{2} + 5x = 0
  2. 2

    Test the candidates until one gives 0. x = 0 is a root. Divide by (x − 0) with synthetic division; the bottom row is the quotient and a 0 remainder.

    01−65000001−650\begin{array}{c|cccc} 0 & 1 & -6 & 5 & 0 \\ & 0 & 0 & 0 & 0 \\ \hline & 1 & -6 & 5 & 0 \end{array}
  3. 3

    What is left is the quadratic x² - 6x + 5 = 0. Solve it (factoring or the quadratic formula):

    x2−6x+5=0⇒x=5, 1x^{2} - 6x + 5 = 0 \Rightarrow x = 5,\ 1
  4. 4

    So the polynomial factors as

    x3−6x2+5x=x(x−1)(x−5)x^{3} - 6x^{2} + 5x = x(x - 1)(x - 5)
LaTeX source of the formulas on this page
\[ x^{3} - 6x^{2} + 5x = x(x - 1)(x - 5) = 0 \]
\[ x^{3} - 6x^{2} + 5x = 0 \]
\[ \begin{array}{c|cccc} 0 & 1 & -6 & 5 & 0 \\ & 0 & 0 & 0 & 0 \\ \hline & 1 & -6 & 5 & 0 \end{array} \]
\[ x^{2} - 6x + 5 = 0 \Rightarrow x = 5,\ 1 \]
\[ x^{3} - 6x^{2} + 5x = x(x - 1)(x - 5) \]

Candidates tested (computed)

Candidate xp(x)Root?
00root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots66
Product of the roots00

Questions

What are the roots of x³ - 6x² + 5x = 0?

The solutions of x³ - 6x² + 5x = 0 are x = 0, 1, 5.

How do you factor x³ - 6x² + 5x?

x(x − 1)(x − 5).

What is the sum of the roots?

6 (it equals −(-6) by Vieta's formulas).

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    Solve x³ - 6x² + 5x = 0: x = 0, 1, 5 | WordGPT