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Expanding and factoringAlgebraHigh schoolabout 1:17

Factor x^4 - 5x^2 + 4: animated lesson

A narrated, step-by-step expanding and factoring video lesson with captions. Final answer: (x - 2)(x - 1)(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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    x4−5x2+4x^{4} - 5x^{2} + 4
  2. 2. Steps, part 1
    1. x4−5x2+4x^{4} - 5x^{2} + 4
      Write the polynomial in standard form
    2. Apply the Rational Root Theorem
    3. x4−5x2+4=(x+2)(x3−2x2−x+2)x^{4} - 5x^{2} + 4 = (x + 2)\left(x^{3} - 2x^{2} - x + 2\right)
      Divide by (x + 2) using synthetic division at x = -2
    4. x3−2x2−x+2=(x+1)(x2−3x+2)x^{3} - 2x^{2} - x + 2 = (x + 1)\left(x^{2} - 3x + 2\right)
      Divide by (x + 1) using synthetic division at x = -1
  3. 3. Steps, part 2
    1. x2−3x+2=(x−1)(x−2)x^{2} - 3x + 2 = (x - 1)\left(x - 2\right)
      Divide by (x - 1) using synthetic division at x = 1
    2. x−2=(x−2)(1)x - 2 = (x - 2)\left(1\right)
      Divide by (x - 2) using synthetic division at x = 2
    3. (x−2)(x−1)(x+1)(x+2)(x - 2)(x - 1)(x + 1)(x + 2)
      Write the complete factorization
  4. 4. The answer
    (x−2)(x−1)(x+1)(x+2)(x - 2)(x - 1)(x + 1)(x + 2)

    x^4 - 5x^2 + 4 = (x - 2)(x - 1)(x + 1)(x + 2)

  5. 5. Check
    • Expanding (x - 2)(x - 1)(x + 1)(x + 2) gives back x^4 - 5x^2 + 4.

Transcript

The problem. Here is the problem: x to the power 4 minus 5x squared plus 4. 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 4 and q dividing the leading coefficient 1. Testing these candidates gives the roots negative 2, negative 1, 1, 2. Divide by (x plus 2) using synthetic division at x equals negative 2. Divide by (x plus 1) using synthetic division at x equals negative 1.

Steps, part 2. Divide by (x minus 1) using synthetic division at x equals 1. Divide by (x minus 2) using synthetic division at x equals 2. Write the complete factorization.

The answer. So the answer is (x minus 2)(x minus 1)(x plus 1)(x plus 2). x to the power 4 minus 5x squared plus 4 factors completely over the rationals as (x minus 2)(x minus 1)(x plus 1)(x plus 2).

Check. Expanding (x minus 2)(x minus 1)(x plus 1)(x plus 2) gives back x to the power 4 minus 5x squared plus 4.

About expanding and factoring

Expand products of brackets and factor polynomials completely.

More lessons like this

Factor x^4 - 5x^2 + 4: Animated Step-by-Step Lesson