Lesson 8 · from §55 of the 1910 text
Sums and Differences of Cubes
Dividing a^3 + b^3 by a + b gives a^2 - ab + b^2, and dividing a^3 - b^3 by a - b gives a^2 + ab + b^2. So a^3 + b^3 = (a + b)(a^2 - ab + b^2), \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Examples. - x^3 + 8 = (x + 2)(x^2 - 2x + 4) - 27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1) - 64x^3 + 125y^6 = (4x + 5y^2)(16x^2 - 20xy^2 + 25y^4)
When an expression is both a difference of cubes and a difference of squares, like a^6 - b^6, treat it as squares first: a^6 - b^6 = (a^3 + b^3)(a^3 - b^3) = (a + b)(a^2 - ab + b^2)(a - b)(a^2 + ab + b^2).
Quick check
y^3 - 27
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Factor completely.
x^3 + 1
x^3 - 1
a^3 - 8
27 + b^3
64x^3 - 1
x^3 - y^6
x^3 + 8y^3
x^3 + y^9
8x^3 - 27y^3
27a^3 + 64b^3
125y^3 + 8x^3
216 - 27x^3
x^6 + y^9
a^6 + b^6
a^6 - b^6
x^6 - 64
64x^6 + 1
1 - 64x^6
a^6b^6 - 64
x^12 + y^12
27x^27 - 1
k^12 - y^6
x^{3a} + 1
x^{3a} - y^{3a}
x^{3a} + y^{6a}
8x^{6a} - y^{9a}
27x^{3m} - 64
Review: factor completely.
x^6 + 2x^3 + 1
x^6 - 2x^3y^3 + y^6
x^4 + x
x^5 - x^2
8x^3 - 64
8x^6 + 216x^3
x^5 - x^3y^2 - x^2y^3 + y^5
x^5 + x^3 + 8x^2 + 8
x^7 - x^4 - 16x^3 + 16
x^6 - 7x^3 - 8
8x^6 - 3 - 23x^3
x^4 + x^3 - x - 1
x^6 - x^2y^2
x^8 + 64x^2
1 + 3x + 3x^2 + x^3
1 - 3x + 3x^2 - x^3
x^3 + 3x^2y + 3xy^2 + y^3
x^3 - 3x^2y + 3xy^2 - y^3
x^3 + 2x^2 + 4x + 8
8 - 4x + 2x^2 - x^3
x^3 + 9x^2 + 27x + 27
x^3 - 9x^2 + 27x - 27
x^3 + 12x^2 + 48x + 64
64 - 48x + 12x^2 - x^3
8x^3 - 12x^2 + 6x - 1
x^4 + x^3 + x^2 - 1
x^3 + a^3 + x^2 - a^2