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Polynomial equationsAlgebraCollegeabout 1:24

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

A narrated, step-by-step polynomial equations video lesson with captions. Final answer: x = 0.63944984, x = 3.5963352; complex: -0.61789252 ± 0.23033172i.

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Solving an equation
    2 x4−6 x3−5 x2+2 x+2=02~{ x}^{4}-6~{ x}^{3}-5~{ x}^{2}+2~ x+2 = 0
  2. 2. Step by step
    1. 2 x4−6 x3−5 x2+2 x+2=02~{ x}^{4}-6~{ x}^{3}-5~{ x}^{2}+2~ x+2 = 0
      Start with the equation
    2. 2x4−6x3−5x2+2x+2=02x^{4} - 6x^{3} - 5x^{2} + 2x + 2 = 0
      Move every term to one side
    3. No further rational roots; approximate the remaining 4 roots numerically
    4. x=0.63944984,x=3.5963352x = 0.63944984,\quad x = 3.5963352
      Collect the real solutions
  3. 3. The answer
    x=0.63944984, x=3.5963352x = 0.63944984,\ x = 3.5963352

    x = 0.63944984, x = 3.5963352

  4. 4. Check
    • x = 0.63944984: left side = 0, right side = 0 ✓
    • x = 3.5963352: left side = 0, right side = 0 ✓

Transcript

The problem. Here is the problem: 2x to the power 4 minus 6x cubed minus 5x squared plus 2x plus 2 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.

The answer. So the answer is x equals 0.63944984, x equals 3.5963352; complex: negative 0.61789252 plus or minus 0.23033172i. 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 0.63944984: left side equals 0, right side equals 0, which checks out. x equals 3.5963352: 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 - 6x^3 - 5x^2 + 2x + 2 = 0: Step-by-Step Lesson