WordGPT

Solve x⁴ - 15x³ + 83x² - 201x + 180 = 0

The solutions of x⁴ - 15x³ + 83x² - 201x + 180 = 0 are x = 3, 3, 4, 5 (repeated roots counted once each: 3, 4, 5).

Explain it step by step in chatMake a short video of it
Graph of y = x⁴ - 15x³ + 83x² - 201x + 1801.522.533.544.555.566.510203040x = 3x = 4x = 5y
Graph of y = x⁴ - 15x³ + 83x² - 201x + 180. The curve crosses the x-axis at x = 3, x = 4, 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.

    x4−15x3+83x2−201x+180=0x^{4} - 15x^{3} + 83x^{2} - 201x + 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.

    31−1583−20118003−36141−1801−1247−600\begin{array}{c|ccccc} 3 & 1 & -15 & 83 & -201 & 180 \\ & 0 & 3 & -36 & 141 & -180 \\ \hline & 1 & -12 & 47 & -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.

    31−1247−6003−27601−9200\begin{array}{c|cccc} 3 & 1 & -12 & 47 & -60 \\ & 0 & 3 & -27 & 60 \\ \hline & 1 & -9 & 20 & 0 \end{array}
  4. 4

    What is left is the quadratic x² - 9x + 20 = 0. Solve it (factoring or the quadratic formula):

    x2−9x+20=0⇒x=5, 4x^{2} - 9x + 20 = 0 \Rightarrow x = 5,\ 4
  5. 5

    So the polynomial factors as

    x4−15x3+83x2−201x+180=(x−3)2(x−4)(x−5)x^{4} - 15x^{3} + 83x^{2} - 201x + 180 = (x - 3)^{2}(x - 4)(x - 5)
LaTeX source of the formulas on this page
\[ x^{4} - 15x^{3} + 83x^{2} - 201x + 180 = (x - 3)^{2}(x - 4)(x - 5) = 0 \]
\[ x^{4} - 15x^{3} + 83x^{2} - 201x + 180 = 0 \]
\[ \begin{array}{c|ccccc} 3 & 1 & -15 & 83 & -201 & 180 \\ & 0 & 3 & -36 & 141 & -180 \\ \hline & 1 & -12 & 47 & -60 & 0 \end{array} \]
\[ \begin{array}{c|cccc} 3 & 1 & -12 & 47 & -60 \\ & 0 & 3 & -27 & 60 \\ \hline & 1 & -9 & 20 & 0 \end{array} \]
\[ x^{2} - 9x + 20 = 0 \Rightarrow x = 5,\ 4 \]
\[ x^{4} - 15x^{3} + 83x^{2} - 201x + 180 = (x - 3)^{2}(x - 4)(x - 5) \]

Candidates tested (computed)

Candidate xp(x)Root?
148no
-1480no
26no
-21050no
30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots1515
Product of the roots180180

Questions

What are the roots of x⁴ - 15x³ + 83x² - 201x + 180 = 0?

The solutions of x⁴ - 15x³ + 83x² - 201x + 180 = 0 are x = 3, 3, 4, 5 (repeated roots counted once each: 3, 4, 5).

How do you factor x⁴ - 15x³ + 83x² - 201x + 180?

(x − 3)(x − 4)(x − 5).

What is the sum of the roots?

15 (it equals −(-15) by Vieta's formulas).

Similar equations

    Solve x⁴ - 15x³ + 83x² - 201x + 180 = 0: x = 3, 4, 5 | WordGPT