WordGPT

Solve x⁴ + 10x³ + 9x² - 160x - 400 = 0

The solutions of x⁴ + 10x³ + 9x² - 160x - 400 = 0 are x = -5, -5, -4, 4 (repeated roots counted once each: -5, -4, 4).

Explain it step by step in chatMake a short video of it
Graph of y = x⁴ + 10x³ + 9x² - 160x - 400-6-4-224-5005001000x = -5x = -4x = 4y
Graph of y = x⁴ + 10x³ + 9x² - 160x - 400. The curve crosses the x-axis at x = -5, x = -4, 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 -400, so the candidates are ±1, ±2, ±4, ±5, ±8, ±10, ±16, ±20, ±25, ±40, ±50, ±80, ±100, ±200, ±400.

    x4+10x3+9x2−160x−400=0x^{4} + 10x^{3} + 9x^{2} - 160x - 400 = 0
  2. 2

    Test the candidates until one gives 0. x = 4 is a root. Divide by (x − 4) with synthetic division; the bottom row is the quotient and a 0 remainder.

    41109−160−4000456260400114651000\begin{array}{c|ccccc} 4 & 1 & 10 & 9 & -160 & -400 \\ & 0 & 4 & 56 & 260 & 400 \\ \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+10x3+9x2−160x−400=(x+5)2(x+4)(x−4)x^{4} + 10x^{3} + 9x^{2} - 160x - 400 = (x + 5)^{2}(x + 4)(x - 4)
LaTeX source of the formulas on this page
\[ x^{4} + 10x^{3} + 9x^{2} - 160x - 400 = (x + 5)^{2}(x + 4)(x - 4) = 0 \]
\[ x^{4} + 10x^{3} + 9x^{2} - 160x - 400 = 0 \]
\[ \begin{array}{c|ccccc} 4 & 1 & 10 & 9 & -160 & -400 \\ & 0 & 4 & 56 & 260 & 400 \\ \hline & 1 & 14 & 65 & 100 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -4 & 1 & 14 & 65 & 100 \\ & 0 & -4 & -40 & -100 \\ \hline & 1 & 10 & 25 & 0 \end{array} \]
\[ x^{2} + 10x + 25 = 0 \Rightarrow x = -5 \]
\[ x^{4} + 10x^{3} + 9x^{2} - 160x - 400 = (x + 5)^{2}(x + 4)(x - 4) \]

Candidates tested (computed)

Candidate xp(x)Root?
1-540no
-1-240no
2-588no
-2-108no
40root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-10-10
Product of the roots-400-400

Questions

What are the roots of x⁴ + 10x³ + 9x² - 160x - 400 = 0?

The solutions of x⁴ + 10x³ + 9x² - 160x - 400 = 0 are x = -5, -5, -4, 4 (repeated roots counted once each: -5, -4, 4).

How do you factor x⁴ + 10x³ + 9x² - 160x - 400?

(x + 5)(x + 4)(x − 4).

What is the sum of the roots?

-10 (it equals −10 by Vieta's formulas).

Similar equations

    Solve x⁴ + 10x³ + 9x² - 160x - 400 = 0: x = -5, -4, 4 | WordGPT