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Solve x⁴ + 14x³ + 73x² + 168x + 144 = 0

The solutions of x⁴ + 14x³ + 73x² + 168x + 144 = 0 are x = -4, -4, -3, -3 (repeated roots counted once each: -4, -3).

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Graph of y = x⁴ + 14x³ + 73x² + 168x + 144-5.5-5-4.5-4-3.5-3-2.5-2-1.52468101214x = -4x = -3y
Graph of y = x⁴ + 14x³ + 73x² + 168x + 144. The curve crosses the x-axis at 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 144, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±16, ±18, ±24, ±36, ±48, ±72, ±144.

    x4+14x3+73x2+168x+144=0x^{4} + 14x^{3} + 73x^{2} + 168x + 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.

    −3114731681440−3−33−120−14411140480\begin{array}{c|ccccc} -3 & 1 & 14 & 73 & 168 & 144 \\ & 0 & -3 & -33 & -120 & -144 \\ \hline & 1 & 11 & 40 & 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.

    −311140480−3−24−4818160\begin{array}{c|cccc} -3 & 1 & 11 & 40 & 48 \\ & 0 & -3 & -24 & -48 \\ \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+14x3+73x2+168x+144=(x+4)2(x+3)2x^{4} + 14x^{3} + 73x^{2} + 168x + 144 = (x + 4)^{2}(x + 3)^{2}
LaTeX source of the formulas on this page
\[ x^{4} + 14x^{3} + 73x^{2} + 168x + 144 = (x + 4)^{2}(x + 3)^{2} = 0 \]
\[ x^{4} + 14x^{3} + 73x^{2} + 168x + 144 = 0 \]
\[ \begin{array}{c|ccccc} -3 & 1 & 14 & 73 & 168 & 144 \\ & 0 & -3 & -33 & -120 & -144 \\ \hline & 1 & 11 & 40 & 48 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -3 & 1 & 11 & 40 & 48 \\ & 0 & -3 & -24 & -48 \\ \hline & 1 & 8 & 16 & 0 \end{array} \]
\[ x^{2} + 8x + 16 = 0 \Rightarrow x = -4 \]
\[ x^{4} + 14x^{3} + 73x^{2} + 168x + 144 = (x + 4)^{2}(x + 3)^{2} \]

Candidates tested (computed)

Candidate xp(x)Root?
1400no
-136no
2900no
-24no
31764no
-30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-14-14
Product of the roots144144

Questions

What are the roots of x⁴ + 14x³ + 73x² + 168x + 144 = 0?

The solutions of x⁴ + 14x³ + 73x² + 168x + 144 = 0 are x = -4, -4, -3, -3 (repeated roots counted once each: -4, -3).

How do you factor x⁴ + 14x³ + 73x² + 168x + 144?

(x + 4)(x + 3).

What is the sum of the roots?

-14 (it equals −14 by Vieta's formulas).

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    Solve x⁴ + 14x³ + 73x² + 168x + 144 = 0: x = -4, -3 | WordGPT