WordGPT

Solve x⁴ - 7x³ - 5x² + 111x - 180 = 0

The solutions of x⁴ - 7x³ - 5x² + 111x - 180 = 0 are x = -4, 3, 3, 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 - 7x³ - 5x² + 111x - 180-4-2246-400-200200400600800x = -4x = 3x = 5y
Graph of y = x⁴ - 7x³ - 5x² + 111x - 180. 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 -180, so the candidates are ±1, ±2, ±3, ±4, ±5, ±6, ±9, ±10, ±12, ±15, ±18, ±20, ±30, ±36, ±45, ±60, ±90, ±180.

    x4−7x3−5x2+111x−180=0x^{4} - 7x^{3} - 5x^{2} + 111x - 180 = 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.

    31−7−5111−18003−12−511801−4−17600\begin{array}{c|ccccc} 3 & 1 & -7 & -5 & 111 & -180 \\ & 0 & 3 & -12 & -51 & 180 \\ \hline & 1 & -4 & -17 & 60 & 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.

    31−4−176003−3−601−1−200\begin{array}{c|cccc} 3 & 1 & -4 & -17 & 60 \\ & 0 & 3 & -3 & -60 \\ \hline & 1 & -1 & -20 & 0 \end{array}
  4. 4

    What is left is the quadratic x² - x - 20 = 0. Solve it (factoring or the quadratic formula):

    x2−x−20=0⇒x=5, −4x^{2} - x - 20 = 0 \Rightarrow x = 5,\ -4
  5. 5

    So the polynomial factors as

    x4−7x3−5x2+111x−180=(x+4)(x−3)2(x−5)x^{4} - 7x^{3} - 5x^{2} + 111x - 180 = (x + 4)(x - 3)^{2}(x - 5)
LaTeX source of the formulas on this page
\[ x^{4} - 7x^{3} - 5x^{2} + 111x - 180 = (x + 4)(x - 3)^{2}(x - 5) = 0 \]
\[ x^{4} - 7x^{3} - 5x^{2} + 111x - 180 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & -7 & -5 & 111 & -180 \\ & 0 & 3 & -12 & -51 & 180 \\ \hline & 1 & -4 & -17 & 60 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 3 & 1 & -4 & -17 & 60 \\ & 0 & 3 & -3 & -60 \\ \hline & 1 & -1 & -20 & 0 \end{array} \]
\[ x^{2} - x - 20 = 0 \Rightarrow x = 5,\ -4 \]
\[ x^{4} - 7x^{3} - 5x^{2} + 111x - 180 = (x + 4)(x - 3)^{2}(x - 5) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-80no
-1-288no
2-18no
-2-350no
30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots77
Product of the roots-180-180

Questions

What are the roots of x⁴ - 7x³ - 5x² + 111x - 180 = 0?

The solutions of x⁴ - 7x³ - 5x² + 111x - 180 = 0 are x = -4, 3, 3, 5 (repeated roots counted once each: -4, 3, 5).

How do you factor x⁴ - 7x³ - 5x² + 111x - 180?

(x + 4)(x − 3)(x − 5).

What is the sum of the roots?

7 (it equals −(-7) by Vieta's formulas).

Similar equations

    Solve x⁴ - 7x³ - 5x² + 111x - 180 = 0: x = -4, 3, 5 | WordGPT