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Expanding and factoringAlgebraHigh schoolabout 1:08

Factor x^3 - 8: animated lesson

A narrated, step-by-step expanding and factoring video lesson with captions. Final answer: (x - 2)(x^2 + 2x + 4).

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 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    x3−8x^{3} - 8
  2. 2. Steps, part 1
    1. x3−8x^{3} - 8
      Write the polynomial in standard form
    2. Recognise a special pattern
    3. Apply the Rational Root Theorem
    4. x3−8=(x−2)(x2+2x+4)x^{3} - 8 = (x - 2)\left(x^{2} + 2x + 4\right)
      Divide by (x - 2) using synthetic division at x = 2
  3. 3. Steps, part 2
    1. Check the remaining quadratic
    2. (x−2)(x2+2x+4)(x - 2)(x^{2} + 2x + 4)
      Write the complete factorization
  4. 4. The answer
    (x−2)(x2+2x+4)(x - 2)(x^{2} + 2x + 4)

    x^3 - 8 = (x - 2)(x^2 + 2x + 4)

  5. 5. Check
    • Expanding (x - 2)(x^2 + 2x + 4) gives back x^3 - 8.

Transcript

The problem. Here is the problem: x cubed minus 8. Factor a polynomial over the rational numbers.

Steps, part 1. Write the polynomial in standard form. Recognise a special pattern. The remaining factor is a difference of cubes: a cubed minus b cubed equals (a minus b)(a squared plus ab plus b squared). Apply the Rational Root Theorem. Any rational root p/q has p dividing the constant term negative 8 and q dividing the leading coefficient 1. Testing these candidates gives the root 2. Divide by (x minus 2) using synthetic division at x equals 2.

Steps, part 2. Check the remaining quadratic. Its discriminant is negative 12, negative, so it is irreducible over the rationals. Write the complete factorization.

The answer. So the answer is (x minus 2)(x squared plus 2x plus 4). x cubed minus 8 factors completely over the rationals as (x minus 2)(x squared plus 2x plus 4).

Check. Expanding (x minus 2)(x squared plus 2x plus 4) gives back x cubed minus 8.

About expanding and factoring

Expand products of brackets and factor polynomials completely.

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Factor x^3 - 8: Expanding and factoring Step-by-Step Lesson