WordGPT

Solve x⁴ - 9x³ + 3x² + 145x - 300 = 0

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

Explain it step by step in chatMake a short video of it
Graph of y = x⁴ - 9x³ + 3x² + 145x - 300-4-2246-600-400-2002004006008001000x = -4x = 3x = 5y
Graph of y = x⁴ - 9x³ + 3x² + 145x - 300. The curve crosses the x-axis at x = -4, x = 3, x = 5, 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.

    31−93145−30003−18−453001−6−151000\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.

    −41−6−151000−440−1001−10250\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+4)(x−3)(x−5)2x^{4} - 9x^{3} + 3x^{2} + 145x - 300 = (x + 4)(x - 3)(x - 5)^{2}
LaTeX source of the formulas on this page
\[ x^{4} - 9x^{3} + 3x^{2} + 145x - 300 = (x + 4)(x - 3)(x - 5)^{2} = 0 \]
\[ x^{4} - 9x^{3} + 3x^{2} + 145x - 300 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & -9 & 3 & 145 & -300 \\ & 0 & 3 & -18 & -45 & 300 \\ \hline & 1 & -6 & -15 & 100 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -4 & 1 & -6 & -15 & 100 \\ & 0 & -4 & 40 & -100 \\ \hline & 1 & -10 & 25 & 0 \end{array} \]
\[ x^{2} - 10x + 25 = 0 \Rightarrow x = 5 \]
\[ x^{4} - 9x^{3} + 3x^{2} + 145x - 300 = (x + 4)(x - 3)(x - 5)^{2} \]

Candidates tested (computed)

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

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots99
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 = -4, 3, 5, 5 (repeated roots counted once each: -4, 3, 5).

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

(x + 4)(x − 3)(x − 5).

What is the sum of the roots?

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

Similar equations

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