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Solve x⁴ + 9x³ + 3x² - 145x - 300 = 0

The solutions of x⁴ + 9x³ + 3x² - 145x - 300 = 0 are x = -5, -5, -3, 4 (repeated roots counted once each: -5, -3, 4).

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Graph of y = x⁴ + 9x³ + 3x² - 145x - 300-6-4-224-600-400-2002004006008001000x = -5x = -3x = 4y
Graph of y = x⁴ + 9x³ + 3x² - 145x - 300. The curve crosses the x-axis at x = -5, 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 -300, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±25, ±30, ±50, ±60, ±75, ±100, ±150, ±300.

    x4+9x3+3x2−145x−300=0x^{4} + 9x^{3} + 3x^{2} - 145x - 300 = 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.

    −3193−145−3000−3−184530016−15−1000\begin{array}{c|ccccc} -3 & 1 & 9 & 3 & -145 & -300 \\ & 0 & -3 & -18 & 45 & 300 \\ \hline & 1 & 6 & -15 & -100 & 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.

    416−15−1000440100110250\begin{array}{c|cccc} 4 & 1 & 6 & -15 & -100 \\ & 0 & 4 & 40 & 100 \\ \hline & 1 & 10 & 25 & 0 \end{array}
  4. 4

    What is left is the quadratic x² + 10x + 25 = 0. Solve it (factoring or the quadratic formula):

    x2+10x+25=0⇒x=−5x^{2} + 10x + 25 = 0 \Rightarrow x = -5
  5. 5

    So the polynomial factors as

    x4+9x3+3x2−145x−300=(x+5)2(x+3)(x−4)x^{4} + 9x^{3} + 3x^{2} - 145x - 300 = (x + 5)^{2}(x + 3)(x - 4)
LaTeX source of the formulas on this page
\[ x^{4} + 9x^{3} + 3x^{2} - 145x - 300 = (x + 5)^{2}(x + 3)(x - 4) = 0 \]
\[ x^{4} + 9x^{3} + 3x^{2} - 145x - 300 = 0 \]
\[ \begin{array}{c|ccccc} -3 & 1 & 9 & 3 & -145 & -300 \\ & 0 & -3 & -18 & 45 & 300 \\ \hline & 1 & 6 & -15 & -100 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 4 & 1 & 6 & -15 & -100 \\ & 0 & 4 & 40 & 100 \\ \hline & 1 & 10 & 25 & 0 \end{array} \]
\[ x^{2} + 10x + 25 = 0 \Rightarrow x = -5 \]
\[ x^{4} + 9x^{3} + 3x^{2} - 145x - 300 = (x + 5)^{2}(x + 3)(x - 4) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-432no
-1-160no
2-490no
-2-54no
3-384no
-30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-9-9
Product of the roots-300-300

Questions

What are the roots of x⁴ + 9x³ + 3x² - 145x - 300 = 0?

The solutions of x⁴ + 9x³ + 3x² - 145x - 300 = 0 are x = -5, -5, -3, 4 (repeated roots counted once each: -5, -3, 4).

How do you factor x⁴ + 9x³ + 3x² - 145x - 300?

(x + 5)(x + 3)(x − 4).

What is the sum of the roots?

-9 (it equals −9 by Vieta's formulas).

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    Solve x⁴ + 9x³ + 3x² - 145x - 300 = 0: x = -5, -3, 4 | WordGPT