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Polynomial equationsAlgebraCollegeabout 1:33

Solve 2x^4 - 5x^3 + 3x^2 - 6x - 6 = 0: animated lesson

A narrated, step-by-step polynomial equations video lesson with captions. Final answer: x = -0.59885094, x = 2.5530165; complex: 0.27291722 ± 1.3739511i.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Solving an equation
    2 x4−5 x3+3 x2−6 x−6=02~{ x}^{4}-5~{ x}^{3}+3~{ x}^{2}-6~ x-6 = 0
  2. 2. Step by step
    1. 2 x4−5 x3+3 x2−6 x−6=02~{ x}^{4}-5~{ x}^{3}+3~{ x}^{2}-6~ x-6 = 0
      Start with the equation
    2. 2x4−5x3+3x2−6x−6=02x^{4} - 5x^{3} + 3x^{2} - 6x - 6 = 0
      Move every term to one side
    3. No further rational roots; approximate the remaining 4 roots numerically
    4. x=−0.59885094,x=2.5530165x = -0.59885094,\quad x = 2.5530165
      Collect the real solutions
  3. 3. On the graph
    -4-3-2-1123456-20-101020304050x = -0.59885094x = 2.5530165
    Each solution is where the graph of left side minus right side crosses zero.
  4. 4. The answer
    x=−0.59885094, x=2.5530165x = -0.59885094,\ x = 2.5530165

    x = -0.59885094, x = 2.5530165

  5. 5. Check
    • x = -0.59885094: left side = 0, right side = 0 ✓
    • x = 2.5530165: left side = 0, right side = 0 ✓

Transcript

The problem. Here is the problem: 2x to the power 4 minus 5x cubed plus 3x squared minus 6x minus 6 equals 0. A polynomial equation of degree 4 in x.

Step by step. Start with the equation. Move every term to one side. No further rational roots; approximate the remaining 4 roots numerically. The Durand–Kerner method iterates on all roots at once until they stop changing. Collect the real solutions.

On the graph. Now look at the graph of the left side minus the right side. It crosses zero exactly at the solutions we found.

The answer. So the answer is x equals negative 0.59885094, x equals 2.5530165; complex: 0.27291722 plus or minus 1.3739511i. A degree-4 polynomial has exactly 4 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

Check. x equals negative 0.59885094: left side equals 0, right side equals 0, which checks out. x equals 2.5530165: left side equals 0, right side equals 0, which checks out.

About polynomial equations

Find all roots of cubic and higher-degree polynomials using the rational root test and factoring.

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Solve 2x^4 - 5x^3 + 3x^2 - 6x - 6 = 0: Step-by-Step Lesson