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Solve x⁴ - 6x³ + 4x² + 6x - 5 = 0

The solutions of x⁴ - 6x³ + 4x² + 6x - 5 = 0 are x = -1, 1, 1, 5 (repeated roots counted once each: -1, 1, 5).

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Graph of y = x⁴ - 6x³ + 4x² + 6x - 5-2-1123456-5050100150200250x = -1x = 1x = 5y
Graph of y = x⁴ - 6x³ + 4x² + 6x - 5. The curve crosses the x-axis at x = -1, x = 1, x = 5, the roots found above.

How to work it out

  1. 1

    A quartic has up to 4 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 -5, so the candidates are ±1, ±5.

    x4−6x3+4x2+6x−5=0x^{4} - 6x^{3} + 4x^{2} + 6x - 5 = 0
  2. 2

    Test the candidates until one gives 0. x = 1 is a root. Divide by (x − 1) with synthetic division; the bottom row is the quotient and a 0 remainder.

    11−646−501−5−151−5−150\begin{array}{c|ccccc} 1 & 1 & -6 & 4 & 6 & -5 \\ & 0 & 1 & -5 & -1 & 5 \\ \hline & 1 & -5 & -1 & 5 & 0 \end{array}
  3. 3

    x = 1 is a root. Divide by (x − 1) with synthetic division; the bottom row is the quotient and a 0 remainder.

    11−5−1501−4−51−4−50\begin{array}{c|cccc} 1 & 1 & -5 & -1 & 5 \\ & 0 & 1 & -4 & -5 \\ \hline & 1 & -4 & -5 & 0 \end{array}
  4. 4

    What is left is the quadratic x² - 4x - 5 = 0. Solve it (factoring or the quadratic formula):

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

    So the polynomial factors as

    x4−6x3+4x2+6x−5=(x+1)(x−1)2(x−5)x^{4} - 6x^{3} + 4x^{2} + 6x - 5 = (x + 1)(x - 1)^{2}(x - 5)
LaTeX source of the formulas on this page
\[ x^{4} - 6x^{3} + 4x^{2} + 6x - 5 = (x + 1)(x - 1)^{2}(x - 5) = 0 \]
\[ x^{4} - 6x^{3} + 4x^{2} + 6x - 5 = 0 \]
\[ \begin{array}{c|ccccc} 1 & 1 & -6 & 4 & 6 & -5 \\ & 0 & 1 & -5 & -1 & 5 \\ \hline & 1 & -5 & -1 & 5 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 1 & 1 & -5 & -1 & 5 \\ & 0 & 1 & -4 & -5 \\ \hline & 1 & -4 & -5 & 0 \end{array} \]
\[ x^{2} - 4x - 5 = 0 \Rightarrow x = 5,\ -1 \]
\[ x^{4} - 6x^{3} + 4x^{2} + 6x - 5 = (x + 1)(x - 1)^{2}(x - 5) \]

Candidates tested (computed)

Candidate xp(x)Root?
10root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots66
Product of the roots-5-5

Questions

What are the roots of x⁴ - 6x³ + 4x² + 6x - 5 = 0?

The solutions of x⁴ - 6x³ + 4x² + 6x - 5 = 0 are x = -1, 1, 1, 5 (repeated roots counted once each: -1, 1, 5).

How do you factor x⁴ - 6x³ + 4x² + 6x - 5?

(x + 1)(x − 1)(x − 5).

What is the sum of the roots?

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

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    Solve x⁴ - 6x³ + 4x² + 6x - 5 = 0: x = -1, 1, 5 | WordGPT