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Solve x⁴ + 3x³ - 25x² - 39x + 180 = 0

The solutions of x⁴ + 3x³ - 25x² - 39x + 180 = 0 are x = -5, -4, 3, 3 (repeated roots counted once each: -5, -4, 3).

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Graph of y = x⁴ + 3x³ - 25x² - 39x + 180-6-4-22450100150200250x = -5x = -4x = 3y
Graph of y = x⁴ + 3x³ - 25x² - 39x + 180. The curve crosses the x-axis at x = -5, x = -4, x = 3, 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+3x3−25x2−39x+180=0x^{4} + 3x^{3} - 25x^{2} - 39x + 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.

    313−25−391800318−21−18016−7−600\begin{array}{c|ccccc} 3 & 1 & 3 & -25 & -39 & 180 \\ & 0 & 3 & 18 & -21 & -180 \\ \hline & 1 & 6 & -7 & -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.

    316−7−6003276019200\begin{array}{c|cccc} 3 & 1 & 6 & -7 & -60 \\ & 0 & 3 & 27 & 60 \\ \hline & 1 & 9 & 20 & 0 \end{array}
  4. 4

    What is left is the quadratic x² + 9x + 20 = 0. Solve it (factoring or the quadratic formula):

    x2+9x+20=0⇒x=−4, −5x^{2} + 9x + 20 = 0 \Rightarrow x = -4,\ -5
  5. 5

    So the polynomial factors as

    x4+3x3−25x2−39x+180=(x+5)(x+4)(x−3)2x^{4} + 3x^{3} - 25x^{2} - 39x + 180 = (x + 5)(x + 4)(x - 3)^{2}
LaTeX source of the formulas on this page
\[ x^{4} + 3x^{3} - 25x^{2} - 39x + 180 = (x + 5)(x + 4)(x - 3)^{2} = 0 \]
\[ x^{4} + 3x^{3} - 25x^{2} - 39x + 180 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & 3 & -25 & -39 & 180 \\ & 0 & 3 & 18 & -21 & -180 \\ \hline & 1 & 6 & -7 & -60 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 3 & 1 & 6 & -7 & -60 \\ & 0 & 3 & 27 & 60 \\ \hline & 1 & 9 & 20 & 0 \end{array} \]
\[ x^{2} + 9x + 20 = 0 \Rightarrow x = -4,\ -5 \]
\[ x^{4} + 3x^{3} - 25x^{2} - 39x + 180 = (x + 5)(x + 4)(x - 3)^{2} \]

Candidates tested (computed)

Candidate xp(x)Root?
1120no
-1192no
242no
-2150no
30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-3-3
Product of the roots180180

Questions

What are the roots of x⁴ + 3x³ - 25x² - 39x + 180 = 0?

The solutions of x⁴ + 3x³ - 25x² - 39x + 180 = 0 are x = -5, -4, 3, 3 (repeated roots counted once each: -5, -4, 3).

How do you factor x⁴ + 3x³ - 25x² - 39x + 180?

(x + 5)(x + 4)(x − 3).

What is the sum of the roots?

-3 (it equals −3 by Vieta's formulas).

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    Solve x⁴ + 3x³ - 25x² - 39x + 180 = 0: x = -5, -4, 3 | WordGPT