WordGPT

Solve x⁴ + 11x³ + 23x² - 95x - 300 = 0

The solutions of x⁴ + 11x³ + 23x² - 95x - 300 = 0 are x = -5, -5, -4, 3 (repeated roots counted once each: -5, -4, 3).

Explain it step by step in chatMake a short video of it
Graph of y = x⁴ + 11x³ + 23x² - 95x - 300-6-4-224-400-200200400600800x = -5x = -4x = 3y
Graph of y = x⁴ + 11x³ + 23x² - 95x - 300. The curve crosses the x-axis at x = -5, 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 -300, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±25, ±30, ±50, ±60, ±75, ±100, ±150, ±300.

    x4+11x3+23x2−95x−300=0x^{4} + 11x^{3} + 23x^{2} - 95x - 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.

    311123−95−3000342195300114651000\begin{array}{c|ccccc} 3 & 1 & 11 & 23 & -95 & -300 \\ & 0 & 3 & 42 & 195 & 300 \\ \hline & 1 & 14 & 65 & 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.

    −4114651000−4−40−100110250\begin{array}{c|cccc} -4 & 1 & 14 & 65 & 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+11x3+23x2−95x−300=(x+5)2(x+4)(x−3)x^{4} + 11x^{3} + 23x^{2} - 95x - 300 = (x + 5)^{2}(x + 4)(x - 3)
LaTeX source of the formulas on this page
\[ x^{4} + 11x^{3} + 23x^{2} - 95x - 300 = (x + 5)^{2}(x + 4)(x - 3) = 0 \]
\[ x^{4} + 11x^{3} + 23x^{2} - 95x - 300 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & 11 & 23 & -95 & -300 \\ & 0 & 3 & 42 & 195 & 300 \\ \hline & 1 & 14 & 65 & 100 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -4 & 1 & 14 & 65 & 100 \\ & 0 & -4 & -40 & -100 \\ \hline & 1 & 10 & 25 & 0 \end{array} \]
\[ x^{2} + 10x + 25 = 0 \Rightarrow x = -5 \]
\[ x^{4} + 11x^{3} + 23x^{2} - 95x - 300 = (x + 5)^{2}(x + 4)(x - 3) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-360no
-1-192no
2-294no
-2-90no
30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-11-11
Product of the roots-300-300

Questions

What are the roots of x⁴ + 11x³ + 23x² - 95x - 300 = 0?

The solutions of x⁴ + 11x³ + 23x² - 95x - 300 = 0 are x = -5, -5, -4, 3 (repeated roots counted once each: -5, -4, 3).

How do you factor x⁴ + 11x³ + 23x² - 95x - 300?

(x + 5)(x + 4)(x − 3).

What is the sum of the roots?

-11 (it equals −11 by Vieta's formulas).

Similar equations

    Solve x⁴ + 11x³ + 23x² - 95x - 300 = 0: x = -5, -4, 3 | WordGPT