EDGE Math 8

Sums or Differences of Two Cubes
A sum or difference of two cubes can be expressed as a product of a binomial and a trinomial as shown below.

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

Example

Factor each polynomial completely.

8 a^{3} + 27

r^{3} t^{3} - 64

b^{3} + 125 c^{6}

Solution

8 a^{3} + 27

= {(2 a)}^{3} + 3^{3}

= (2 a + 3)

[{(2 a)}^{2} - (2 a) (3) + {(3)}^{2}]

= (2 a + 3) (4 a^{2} - 6 a + 9)

r^{3} t^{3} - 64

= {(r t)}^{3} - 4^{3}

= (r t - 4)

[{(r t)}^{2} + (r t) (4) + 4^{2}]

= (r t - 4) (r^{2} t^{2} + 4 r t + 16)

b^{3} + 125 c^{6}

= b^{3} + {(5 c^{2})}^{3}

= (b + 5 c^{2})

[b^{2} - (b) (5 c^{2}) + {(5 c^{2})}^{2}]

= (b + 5 c^{2}) (b^{2} - 5 b c^{2} + 25 c^{4})