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Solve x⁴ + 17x³ + 108x² + 304x + 320 = 0

The solutions of x⁴ + 17x³ + 108x² + 304x + 320 = 0 are x = -5, -4, -4, -4 (repeated roots counted once each: -5, -4).

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Graph of y = x⁴ + 17x³ + 108x² + 304x + 320-6.5-6-5.5-5-4.5-4-3.5-3-2.55101520x = -5x = -4y
Graph of y = x⁴ + 17x³ + 108x² + 304x + 320. The curve crosses the x-axis at x = -5, 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 320, so the candidates are ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±32, ±40, ±64, ±80, ±160, ±320.

    x4+17x3+108x2+304x+320=0x^{4} + 17x^{3} + 108x^{2} + 304x + 320 = 0
  2. 2

    Test the candidates until one gives 0. x = -4 is a root. Divide by (x + 4) with synthetic division; the bottom row is the quotient and a 0 remainder.

    −41171083043200−4−52−224−32011356800\begin{array}{c|ccccc} -4 & 1 & 17 & 108 & 304 & 320 \\ & 0 & -4 & -52 & -224 & -320 \\ \hline & 1 & 13 & 56 & 80 & 0 \end{array}
  3. 3

    x = -4 is a root. Divide by (x + 4) with synthetic division; the bottom row is the quotient and a 0 remainder.

    −411356800−4−36−8019200\begin{array}{c|cccc} -4 & 1 & 13 & 56 & 80 \\ & 0 & -4 & -36 & -80 \\ \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+17x3+108x2+304x+320=(x+5)(x+4)3x^{4} + 17x^{3} + 108x^{2} + 304x + 320 = (x + 5)(x + 4)^{3}
LaTeX source of the formulas on this page
\[ x^{4} + 17x^{3} + 108x^{2} + 304x + 320 = (x + 5)(x + 4)^{3} = 0 \]
\[ x^{4} + 17x^{3} + 108x^{2} + 304x + 320 = 0 \]
\[ \begin{array}{c|ccccc} -4 & 1 & 17 & 108 & 304 & 320 \\ & 0 & -4 & -52 & -224 & -320 \\ \hline & 1 & 13 & 56 & 80 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -4 & 1 & 13 & 56 & 80 \\ & 0 & -4 & -36 & -80 \\ \hline & 1 & 9 & 20 & 0 \end{array} \]
\[ x^{2} + 9x + 20 = 0 \Rightarrow x = -4,\ -5 \]
\[ x^{4} + 17x^{3} + 108x^{2} + 304x + 320 = (x + 5)(x + 4)^{3} \]

Candidates tested (computed)

Candidate xp(x)Root?
1750no
-1108no
21512no
-224no
44608no
-40root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-17-17
Product of the roots320320

Questions

What are the roots of x⁴ + 17x³ + 108x² + 304x + 320 = 0?

The solutions of x⁴ + 17x³ + 108x² + 304x + 320 = 0 are x = -5, -4, -4, -4 (repeated roots counted once each: -5, -4).

How do you factor x⁴ + 17x³ + 108x² + 304x + 320?

(x + 5)(x + 4).

What is the sum of the roots?

-17 (it equals −17 by Vieta's formulas).

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    Solve x⁴ + 17x³ + 108x² + 304x + 320 = 0: x = -5, -4 | WordGPT