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Solve x⁴ + 6x³ - 7x² - 96x - 144 = 0

The solutions of x⁴ + 6x³ - 7x² - 96x - 144 = 0 are x = -4, -3, -3, 4 (repeated roots counted once each: -4, -3, 4).

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Graph of y = x⁴ + 6x³ - 7x² - 96x - 144-4-224-400-200200400600800x = -4x = -3x = 4y
Graph of y = x⁴ + 6x³ - 7x² - 96x - 144. 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 -144, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±16, ±18, ±24, ±36, ±48, ±72, ±144.

    x4+6x3−7x2−96x−144=0x^{4} + 6x^{3} - 7x^{2} - 96x - 144 = 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.

    −316−7−96−1440−3−94814413−16−480\begin{array}{c|ccccc} -3 & 1 & 6 & -7 & -96 & -144 \\ & 0 & -3 & -9 & 48 & 144 \\ \hline & 1 & 3 & -16 & -48 & 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.

    −313−16−480−304810−160\begin{array}{c|cccc} -3 & 1 & 3 & -16 & -48 \\ & 0 & -3 & 0 & 48 \\ \hline & 1 & 0 & -16 & 0 \end{array}
  4. 4

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

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

    So the polynomial factors as

    x4+6x3−7x2−96x−144=(x+4)(x+3)2(x−4)x^{4} + 6x^{3} - 7x^{2} - 96x - 144 = (x + 4)(x + 3)^{2}(x - 4)
LaTeX source of the formulas on this page
\[ x^{4} + 6x^{3} - 7x^{2} - 96x - 144 = (x + 4)(x + 3)^{2}(x - 4) = 0 \]
\[ x^{4} + 6x^{3} - 7x^{2} - 96x - 144 = 0 \]
\[ \begin{array}{c|ccccc} -3 & 1 & 6 & -7 & -96 & -144 \\ & 0 & -3 & -9 & 48 & 144 \\ \hline & 1 & 3 & -16 & -48 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -3 & 1 & 3 & -16 & -48 \\ & 0 & -3 & 0 & 48 \\ \hline & 1 & 0 & -16 & 0 \end{array} \]
\[ x^{2} - 16 = 0 \Rightarrow x = 4,\ -4 \]
\[ x^{4} + 6x^{3} - 7x^{2} - 96x - 144 = (x + 4)(x + 3)^{2}(x - 4) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-240no
-1-60no
2-300no
-2-12no
3-252no
-30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-6-6
Product of the roots-144-144

Questions

What are the roots of x⁴ + 6x³ - 7x² - 96x - 144 = 0?

The solutions of x⁴ + 6x³ - 7x² - 96x - 144 = 0 are x = -4, -3, -3, 4 (repeated roots counted once each: -4, -3, 4).

How do you factor x⁴ + 6x³ - 7x² - 96x - 144?

(x + 4)(x + 3)(x − 4).

What is the sum of the roots?

-6 (it equals −6 by Vieta's formulas).

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    Solve x⁴ + 6x³ - 7x² - 96x - 144 = 0: x = -4, -3, 4 | WordGPT