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Expanding and factoringAlgebraHigh schoolabout 1:10

Factor x^3 - 3x^2 - 4x + 12: animated lesson

A narrated, step-by-step expanding and factoring video lesson with captions. Final answer: (x - 3)(x - 2)(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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    x3−3x2−4x+12x^{3} - 3x^{2} - 4x + 12
  2. 2. Steps, part 1
    1. x3−3x2−4x+12x^{3} - 3x^{2} - 4x + 12
      Write the polynomial in standard form
    2. Apply the Rational Root Theorem
    3. x3−3x2−4x+12=(x+2)(x2−5x+6)x^{3} - 3x^{2} - 4x + 12 = (x + 2)\left(x^{2} - 5x + 6\right)
      Divide by (x + 2) using synthetic division at x = -2
    4. x2−5x+6=(x−2)(x−3)x^{2} - 5x + 6 = (x - 2)\left(x - 3\right)
      Divide by (x - 2) using synthetic division at x = 2
  3. 3. Steps, part 2
    1. x−3=(x−3)(1)x - 3 = (x - 3)\left(1\right)
      Divide by (x - 3) using synthetic division at x = 3
    2. (x−3)(x−2)(x+2)(x - 3)(x - 2)(x + 2)
      Write the complete factorization
  4. 4. The answer
    (x−3)(x−2)(x+2)(x - 3)(x - 2)(x + 2)

    x^3 - 3x^2 - 4x + 12 = (x - 3)(x - 2)(x + 2)

  5. 5. Check
    • Expanding (x - 3)(x - 2)(x + 2) gives back x^3 - 3x^2 - 4x + 12.

Transcript

The problem. Here is the problem: x cubed minus 3x squared minus 4x plus 12. Factor a polynomial over the rational numbers.

Steps, part 1. Write the polynomial in standard form. Apply the Rational Root Theorem. Any rational root p/q has p dividing the constant term 12 and q dividing the leading coefficient 1. Testing these candidates gives the roots negative 2, 2, 3. Divide by (x plus 2) using synthetic division at x equals negative 2. Divide by (x minus 2) using synthetic division at x equals 2.

Steps, part 2. Divide by (x minus 3) using synthetic division at x equals 3. Write the complete factorization.

The answer. So the answer is (x minus 3)(x minus 2)(x plus 2). x cubed minus 3x squared minus 4x plus 12 factors completely over the rationals as (x minus 3)(x minus 2)(x plus 2).

Check. Expanding (x minus 3)(x minus 2)(x plus 2) gives back x cubed minus 3x squared minus 4x plus 12.

About expanding and factoring

Expand products of brackets and factor polynomials completely.

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Factor x^3 - 3x^2 - 4x + 12: Animated Step-by-Step Lesson