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Solve x⁴ + 16x³ + 95x² + 248x + 240 = 0

The solutions of x⁴ + 16x³ + 95x² + 248x + 240 = 0 are x = -5, -4, -4, -3 (repeated roots counted once each: -5, -4, -3).

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Graph of y = x⁴ + 16x³ + 95x² + 248x + 240-6.5-6-5.5-5-4.5-4-3.5-3-2.5-2-1.551015202530x = -5x = -4x = -3y
Graph of y = x⁴ + 16x³ + 95x² + 248x + 240. 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 240, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±16, ±20, ±24, ±30, ±40, ±48, ±60, ±80, ±120, ±240.

    x4+16x3+95x2+248x+240=0x^{4} + 16x^{3} + 95x^{2} + 248x + 240 = 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.

    −3116952482400−3−39−168−24011356800\begin{array}{c|ccccc} -3 & 1 & 16 & 95 & 248 & 240 \\ & 0 & -3 & -39 & -168 & -240 \\ \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+16x3+95x2+248x+240=(x+5)(x+4)2(x+3)x^{4} + 16x^{3} + 95x^{2} + 248x + 240 = (x + 5)(x + 4)^{2}(x + 3)
LaTeX source of the formulas on this page
\[ x^{4} + 16x^{3} + 95x^{2} + 248x + 240 = (x + 5)(x + 4)^{2}(x + 3) = 0 \]
\[ x^{4} + 16x^{3} + 95x^{2} + 248x + 240 = 0 \]
\[ \begin{array}{c|ccccc} -3 & 1 & 16 & 95 & 248 & 240 \\ & 0 & -3 & -39 & -168 & -240 \\ \hline & 1 & 13 & 56 & 80 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -4 & 1 & 13 & 56 & 80 \\ & 0 & -4 & -36 & -80 \\ \hline & 1 & 9 & 20 & 0 \end{array} \]
\[ x^{2} + 9x + 20 = 0 \Rightarrow x = -4,\ -5 \]
\[ x^{4} + 16x^{3} + 95x^{2} + 248x + 240 = (x + 5)(x + 4)^{2}(x + 3) \]

Candidates tested (computed)

Candidate xp(x)Root?
1600no
-172no
21260no
-212no
32352no
-30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-16-16
Product of the roots240240

Questions

What are the roots of x⁴ + 16x³ + 95x² + 248x + 240 = 0?

The solutions of x⁴ + 16x³ + 95x² + 248x + 240 = 0 are x = -5, -4, -4, -3 (repeated roots counted once each: -5, -4, -3).

How do you factor x⁴ + 16x³ + 95x² + 248x + 240?

(x + 5)(x + 4)(x + 3).

What is the sum of the roots?

-16 (it equals −16 by Vieta's formulas).

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    Solve x⁴ + 16x³ + 95x² + 248x + 240 = 0: x = -5, -4, -3 | WordGPT