WordGPT

Solve x⁴ + 15x³ + 84x² + 208x + 192 = 0

The solutions of x⁴ + 15x³ + 84x² + 208x + 192 = 0 are x = -4, -4, -4, -3 (repeated roots counted once each: -4, -3).

Explain it step by step in chatMake a short video of it
Graph of y = x⁴ + 15x³ + 84x² + 208x + 192-5.5-5-4.5-4-3.5-3-2.5-2-1.55101520x = -4x = -3y
Graph of y = x⁴ + 15x³ + 84x² + 208x + 192. The curve crosses the x-axis at 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 192, so the candidates are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±32, ±48, ±64, ±96, ±192.

    x4+15x3+84x2+208x+192=0x^{4} + 15x^{3} + 84x^{2} + 208x + 192 = 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.

    −3115842081920−3−36−144−19211248640\begin{array}{c|ccccc} -3 & 1 & 15 & 84 & 208 & 192 \\ & 0 & -3 & -36 & -144 & -192 \\ \hline & 1 & 12 & 48 & 64 & 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.

    −411248640−4−32−6418160\begin{array}{c|cccc} -4 & 1 & 12 & 48 & 64 \\ & 0 & -4 & -32 & -64 \\ \hline & 1 & 8 & 16 & 0 \end{array}
  4. 4

    What is left is the quadratic x² + 8x + 16 = 0. Solve it (factoring or the quadratic formula):

    x2+8x+16=0⇒x=−4x^{2} + 8x + 16 = 0 \Rightarrow x = -4
  5. 5

    So the polynomial factors as

    x4+15x3+84x2+208x+192=(x+4)3(x+3)x^{4} + 15x^{3} + 84x^{2} + 208x + 192 = (x + 4)^{3}(x + 3)
LaTeX source of the formulas on this page
\[ x^{4} + 15x^{3} + 84x^{2} + 208x + 192 = (x + 4)^{3}(x + 3) = 0 \]
\[ x^{4} + 15x^{3} + 84x^{2} + 208x + 192 = 0 \]
\[ \begin{array}{c|ccccc} -3 & 1 & 15 & 84 & 208 & 192 \\ & 0 & -3 & -36 & -144 & -192 \\ \hline & 1 & 12 & 48 & 64 & 0 \end{array} \]
\[ \begin{array}{c|cccc} -4 & 1 & 12 & 48 & 64 \\ & 0 & -4 & -32 & -64 \\ \hline & 1 & 8 & 16 & 0 \end{array} \]
\[ x^{2} + 8x + 16 = 0 \Rightarrow x = -4 \]
\[ x^{4} + 15x^{3} + 84x^{2} + 208x + 192 = (x + 4)^{3}(x + 3) \]

Candidates tested (computed)

Candidate xp(x)Root?
1500no
-154no
21080no
-28no
32058no
-30root

Vieta's formulas check the answer (computed)

RelationFrom the rootsFrom the coefficients
Sum of the roots-15-15
Product of the roots192192

Questions

What are the roots of x⁴ + 15x³ + 84x² + 208x + 192 = 0?

The solutions of x⁴ + 15x³ + 84x² + 208x + 192 = 0 are x = -4, -4, -4, -3 (repeated roots counted once each: -4, -3).

How do you factor x⁴ + 15x³ + 84x² + 208x + 192?

(x + 4)(x + 3).

What is the sum of the roots?

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

Similar equations

    Solve x⁴ + 15x³ + 84x² + 208x + 192 = 0: x = -4, -3 | WordGPT