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Polynomial equationsAlgebraHigh schoolabout 2:02

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

A narrated, step-by-step polynomial equations video lesson with captions. Final answer: x = -2, x = -1, x = 0, x = 1, x = 2.

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 (0.8 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Solving an equation
    x5−5 x3+4 x=0{ x}^{5}-5~{ x}^{3}+4~ x = 0
  2. 2. Steps, part 1
    1. x5−5 x3+4 x=0{ x}^{5}-5~{ x}^{3}+4~ x = 0
      Start with the equation
    2. x5−5x3+4x=0x^{5} - 5x^{3} + 4x = 0
      Move every term to one side
    3. x4−5x2+4=0x^{4} - 5x^{2} + 4 = 0
      Test candidates from the rational root theorem: x = 0 is a root
    4. x3−2x2−x+2=0x^{3} - 2x^{2} - x + 2 = 0
      Test candidates from the rational root theorem: x = -2 is a root
  3. 3. Steps, part 2
    1. x2−3x+2=0x^{2} - 3x + 2 = 0
      Test candidates from the rational root theorem: x = -1 is a root
    2. x−2=0x - 2 = 0
      Test candidates from the rational root theorem: x = 1 is a root
    3. Test candidates from the rational root theorem: x = 2 is a root
    4. x=−2,x=−1,x=0,x=1,x=2x = -2,\quad x = -1,\quad x = 0,\quad x = 1,\quad x = 2
      Collect the real solutions
  4. 4. The answer
    x=−2, x=−1, x=0, x=1, x=2x = -2,\ x = -1,\ x = 0,\ x = 1,\ x = 2

    x = -2, x = -1, x = 0, x = 1, x = 2

  5. 5. Check
    • x = -2: left side = 0, right side = 0 ✓
    • x = -1: left side = 0, right side = 0 ✓
    • x = 0: left side = 0, right side = 0 ✓
    • x = 1: left side = 0, right side = 0 ✓

Transcript

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

Steps, part 1. Start with the equation. Move every term to one side. Test candidates from the rational root theorem: x equals 0 is a root. Synthetic division by (x minus 0) leaves a polynomial of degree 4. Test candidates from the rational root theorem: x equals negative 2 is a root. Synthetic division by (x plus 2) leaves a polynomial of degree 3.

Steps, part 2. Test candidates from the rational root theorem: x equals negative 1 is a root. Synthetic division by (x plus 1) leaves a polynomial of degree 2. Test candidates from the rational root theorem: x equals 1 is a root. Synthetic division by (x minus 1) leaves a polynomial of degree 1. Test candidates from the rational root theorem: x equals 2 is a root. Synthetic division by (x minus 2) leaves a polynomial of degree 0. Collect the real solutions.

The answer. So the answer is x equals negative 2, x equals negative 1, x equals 0, x equals 1, x equals 2. A degree-5 polynomial has exactly 5 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 2: left side equals 0, right side equals 0, which checks out. x equals negative 1: left side equals 0, right side equals 0, which checks out. x equals 0: left side equals 0, right side equals 0, which checks out. x equals 1: 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 x^5 - 5x^3 + 4x = 0: Animated Step-by-Step Lesson