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Solve x⁴ - 5x³ - 17x² + 129x - 180 = 0

The solutions of x⁴ - 5x³ - 17x² + 129x - 180 = 0 are x = -5, 3, 3, 4 (repeated roots counted once each: -5, 3, 4).

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Graph of y = x⁴ - 5x³ - 17x² + 129x - 180-6-4-224-600-400-20020040060080010001200x = -5x = 3x = 4y
Graph of y = x⁴ - 5x³ - 17x² + 129x - 180. The curve crosses the x-axis at x = -5, x = 3, x = 4, 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 -180, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±9, ±10, ±12, ±15, ±18, ±20, ±30, ±36, ±45, ±60, ±90, ±180.

    x4−5x3−17x2+129x−180=0x^{4} - 5x^{3} - 17x^{2} + 129x - 180 = 0
  2. 2

    Test the candidates until one gives 0. x = 3 is a root. Divide by (x − 3) with synthetic division; the bottom row is the quotient and a 0 remainder.

    31−5−17129−18003−6−691801−2−23600\begin{array}{c|ccccc} 3 & 1 & -5 & -17 & 129 & -180 \\ & 0 & 3 & -6 & -69 & 180 \\ \hline & 1 & -2 & -23 & 60 & 0 \end{array}
  3. 3

    x = 3 is a root. Divide by (x − 3) with synthetic division; the bottom row is the quotient and a 0 remainder.

    31−2−2360033−6011−200\begin{array}{c|cccc} 3 & 1 & -2 & -23 & 60 \\ & 0 & 3 & 3 & -60 \\ \hline & 1 & 1 & -20 & 0 \end{array}
  4. 4

    What is left is the quadratic x² + x - 20 = 0. Solve it (factoring or the quadratic formula):

    x2+x−20=0⇒x=4, −5x^{2} + x - 20 = 0 \Rightarrow x = 4,\ -5
  5. 5

    So the polynomial factors as

    x4−5x3−17x2+129x−180=(x+5)(x−3)2(x−4)x^{4} - 5x^{3} - 17x^{2} + 129x - 180 = (x + 5)(x - 3)^{2}(x - 4)
LaTeX source of the formulas on this page
\[ x^{4} - 5x^{3} - 17x^{2} + 129x - 180 = (x + 5)(x - 3)^{2}(x - 4) = 0 \]
\[ x^{4} - 5x^{3} - 17x^{2} + 129x - 180 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & -5 & -17 & 129 & -180 \\ & 0 & 3 & -6 & -69 & 180 \\ \hline & 1 & -2 & -23 & 60 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 3 & 1 & -2 & -23 & 60 \\ & 0 & 3 & 3 & -60 \\ \hline & 1 & 1 & -20 & 0 \end{array} \]
\[ x^{2} + x - 20 = 0 \Rightarrow x = 4,\ -5 \]
\[ x^{4} - 5x^{3} - 17x^{2} + 129x - 180 = (x + 5)(x - 3)^{2}(x - 4) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-72no
-1-320no
2-14no
-2-450no
30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots55
Product of the roots-180-180

Questions

What are the roots of x⁴ - 5x³ - 17x² + 129x - 180 = 0?

The solutions of x⁴ - 5x³ - 17x² + 129x - 180 = 0 are x = -5, 3, 3, 4 (repeated roots counted once each: -5, 3, 4).

How do you factor x⁴ - 5x³ - 17x² + 129x - 180?

(x + 5)(x − 3)(x − 4).

What is the sum of the roots?

5 (it equals −(-5) by Vieta's formulas).

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    Solve x⁴ - 5x³ - 17x² + 129x - 180 = 0: x = -5, 3, 4 | WordGPT