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Polynomial equationsAlgebraHigh schoolabout 1:22

Solve x^3 + 1 = 0: animated lesson

A narrated, step-by-step polynomial equations video lesson with captions. Final answer: x = -1; complex: 0.5 ± 0.8660254i.

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
    x3+1=0{ x}^{3}+1 = 0
  2. 2. Steps, part 1
    1. x3+1=0{ x}^{3}+1 = 0
      Start with the equation
    2. x3+1=0x^{3} + 1 = 0
      Move every term to one side
    3. x2−x+1=0x^{2} - x + 1 = 0
      Test candidates from the rational root theorem: x = -1 is a root
    4. x2−x+1=0x^{2} - x + 1 = 0
      Solve the remaining quadratic
  3. 3. Steps, part 2
    1. Δ=b2−4ac=(−1)2−4⋅1⋅1=−3\Delta = b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot 1 = -3
      Compute the discriminant
    2. Δ < 0, so there are no real roots; the two roots are complex conjugates
    3. x=1±−32=1±3 i2x = \frac{1 \pm \sqrt{-3}}{2} = \frac{1 \pm \sqrt{3}\,i}{2}
      Apply the quadratic formula with √(−1) = i
    4. x=1±3 i2x = \frac{1 \pm \sqrt{3}\,i}{2}
      Simplify
  4. 4. Steps, part 3
    1. x=−1x = -1
      Collect the real solutions
  5. 5. On the graph
    -4-3-2-112-50-40-30-20-1010x = -1
    Each solution is where the graph of left side minus right side crosses zero.
  6. 6. The answer
    x=−1x = -1

    x = -1

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

Transcript

The problem. Here is the problem: x cubed plus 1 equals 0. A polynomial equation of degree 3 in x.

Steps, part 1. Start with the equation. Move every term to one side. 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. Solve the remaining quadratic.

Steps, part 2. Compute the discriminant. delta less than 0, so there are no real roots; the two roots are complex conjugates. Apply the quadratic formula with the square root of negative 1 equals i. Simplify.

Steps, part 3. 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 1; complex: 0.5 plus or minus 0.8660254i. A degree-3 polynomial has exactly 3 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 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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