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Solve x⁴ + 7x³ - 4x² - 112x - 192 = 0

The solutions of x⁴ + 7x³ - 4x² - 112x - 192 = 0 are x = -4, -4, -3, 4 (repeated roots counted once each: -4, -3, 4).

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Graph of y = x⁴ + 7x³ - 4x² - 112x - 192-4-224-400-200200400600800x = -4x = -3x = 4y
Graph of y = x⁴ + 7x³ - 4x² - 112x - 192. The curve crosses the x-axis at x = -4, 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 -192, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±32, ±48, ±64, ±96, ±192.

    x4+7x3−4x2−112x−192=0x^{4} + 7x^{3} - 4x^{2} - 112x - 192 = 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.

    −317−4−112−1920−3−124819214−16−640\begin{array}{c|ccccc} -3 & 1 & 7 & -4 & -112 & -192 \\ & 0 & -3 & -12 & 48 & 192 \\ \hline & 1 & 4 & -16 & -64 & 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.

    414−16−6404326418160\begin{array}{c|cccc} 4 & 1 & 4 & -16 & -64 \\ & 0 & 4 & 32 & 64 \\ \hline & 1 & 8 & 16 & 0 \end{array}
  4. 4

    What is left is the quadratic x² + 8x + 16 = 0. Solve it (factoring or the quadratic formula):

    x2+8x+16=0⇒x=−4x^{2} + 8x + 16 = 0 \Rightarrow x = -4
  5. 5

    So the polynomial factors as

    x4+7x3−4x2−112x−192=(x+4)2(x+3)(x−4)x^{4} + 7x^{3} - 4x^{2} - 112x - 192 = (x + 4)^{2}(x + 3)(x - 4)
LaTeX source of the formulas on this page
\[ x^{4} + 7x^{3} - 4x^{2} - 112x - 192 = (x + 4)^{2}(x + 3)(x - 4) = 0 \]
\[ x^{4} + 7x^{3} - 4x^{2} - 112x - 192 = 0 \]
\[ \begin{array}{c|ccccc} -3 & 1 & 7 & -4 & -112 & -192 \\ & 0 & -3 & -12 & 48 & 192 \\ \hline & 1 & 4 & -16 & -64 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 4 & 1 & 4 & -16 & -64 \\ & 0 & 4 & 32 & 64 \\ \hline & 1 & 8 & 16 & 0 \end{array} \]
\[ x^{2} + 8x + 16 = 0 \Rightarrow x = -4 \]
\[ x^{4} + 7x^{3} - 4x^{2} - 112x - 192 = (x + 4)^{2}(x + 3)(x - 4) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-300no
-1-90no
2-360no
-2-24no
3-294no
-30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-7-7
Product of the roots-192-192

Questions

What are the roots of x⁴ + 7x³ - 4x² - 112x - 192 = 0?

The solutions of x⁴ + 7x³ - 4x² - 112x - 192 = 0 are x = -4, -4, -3, 4 (repeated roots counted once each: -4, -3, 4).

How do you factor x⁴ + 7x³ - 4x² - 112x - 192?

(x + 4)(x + 3)(x − 4).

What is the sum of the roots?

-7 (it equals −7 by Vieta's formulas).

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    Solve x⁴ + 7x³ - 4x² - 112x - 192 = 0: x = -4, -3, 4 | WordGPT