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Expanding and factoringAlgebraHigh schoolabout 1:18

Factor x^4 - 81: animated lesson

A narrated, step-by-step expanding and factoring video lesson with captions. Final answer: (x - 3)(x + 3)(x^2 + 9).

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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. The problem
Worked example
$x^{4} - 81$
2. Steps, part 1
1. $x^{4} - 81$
  Write the polynomial in standard form
2. Recognise a special pattern
3. Apply the Rational Root Theorem
4. $x^{4} - 81 = (x + 3)\left(x^{3} - 3x^{2} + 9x - 27\right)$
  Divide by (x + 3) using synthetic division at x = -3
3. Steps, part 2
1. $x^{3} - 3x^{2} + 9x - 27 = (x - 3)\left(x^{2} + 9\right)$
  Divide by (x - 3) using synthetic division at x = 3
2. Check the remaining quadratic
3. $(x - 3)(x + 3)(x^{2} + 9)$
  Write the complete factorization
4. The answer
$(x - 3)(x + 3)(x^{2} + 9)$
x^4 - 81 = (x - 3)(x + 3)(x^2 + 9)
5. Check
- Expanding (x - 3)(x + 3)(x^2 + 9) gives back x^4 - 81.

Transcript

The problem. Here is the problem: x to the power 4 minus 81. 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 squares: a squared minus b squared equals (a minus b)(a plus b). Apply the Rational Root Theorem. Any rational root p/q has p dividing the constant term negative 81 and q dividing the leading coefficient 1. Testing these candidates gives the roots negative 3, 3. Divide by (x plus 3) using synthetic division at x equals negative 3.

Steps, part 2. Divide by (x minus 3) using synthetic division at x equals 3. Check the remaining quadratic. Its discriminant is negative 36, negative, so it is irreducible over the rationals. Write the complete factorization.

The answer. So the answer is (x minus 3)(x plus 3)(x squared plus 9). x to the power 4 minus 81 factors completely over the rationals as (x minus 3)(x plus 3)(x squared plus 9).

Check. Expanding (x minus 3)(x plus 3)(x squared plus 9) gives back x to the power 4 minus 81.

About expanding and factoring

Expand products of brackets and factor polynomials completely.

Factor x^4 - 81: Expanding and factoring Step-by-Step Lesson