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Expanding and factoringAlgebraHigh schoolabout 1:15

Factor 8x^3 - 27: animated lesson

A narrated, step-by-step expanding and factoring video lesson with captions. Final answer: (2x - 3)(4x^2 + 6x + 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
$8x^{3} - 27$
2. Steps, part 1
1. $8x^{3} - 27$
  Write the polynomial in standard form
2. Recognise a special pattern
3. Apply the Rational Root Theorem
4. $8x^{3} - 27 = (2x - 3)\left(4x^{2} + 6x + 9\right)$
  Divide by (2x - 3) using synthetic division at x = 3/2
3. Steps, part 2
1. Check the remaining quadratic
2. $(2x - 3)(4x^{2} + 6x + 9)$
  Write the complete factorization
4. The answer
$(2x - 3)(4x^{2} + 6x + 9)$
8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9)
5. Check
- Expanding (2x - 3)(4x^2 + 6x + 9) gives back 8x^3 - 27.

Transcript

The problem. Here is the problem: 8x cubed minus 27. 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 27 and q dividing the leading coefficient 8. Testing these candidates gives the root 3 over 2. Divide by (2x minus 3) using synthetic division at x equals 3 over 2.

Steps, part 2. Check the remaining quadratic. Its discriminant is negative 108, negative, so it is irreducible over the rationals. Write the complete factorization.

The answer. So the answer is (2x minus 3)(4x squared plus 6x plus 9). 8x cubed minus 27 factors completely over the rationals as (2x minus 3)(4x squared plus 6x plus 9).

Check. Expanding (2x minus 3)(4x squared plus 6x plus 9) gives back 8x cubed minus 27.

About expanding and factoring

Expand products of brackets and factor polynomials completely.

Factor 8x^3 - 27: Animated Step-by-Step Lesson (High school)