## FACTOR SUMS AND DIFFERENCES OF CUBES

Factor sums and differences of cubes :

Before going to see the example problems first we have to know about the formula for a³ - b³ and a³ + b³

a³ - b³ = (a - b) (a² + ab + b²)

a³ + b³ = (a + b) (a² -  ab + b²)

Whenever we have expression like a³ - b³ and a³ + b³, we can expand this using the above formula.

## Factor sums and differences of cubes - Examples

Example 1 :

Factorize  8 x³ + 125 y³

Solution :

x³ + 125 y³  =  2³ x³ + 5³ y³ ==> (2 x)³ + (5 y)³

We can compare the above question with the formula a³+ b³. From this we have to expand this using the formula (a + b) (a² - ab + b²).

Instead of "a" we have "2x" and instead of "b" we have "5y"

=  (2x + 5y) [(2x)² + (2x) (5y) + (5y)²]

=  (2x + 5y) (4 x² + 10 xy + 25 y²)

Example 2 :

Factorize  27 x³ - 64 y³

Solution :

27 x³ - 64 y³  =  3³ x³ - 4³ y³ ==> (3 x)³ - (4 y)³

We can compare the above question with the formula a³- b³. From this we have to expand this using the formula (a - b) (a² + ab + b²).

Instead of "a" we have "3x" and instead of "b" we have "4y"

=  (3x - 4y) [(3x)² + (3x) (4y) + (4y)²]

=  (3x - 4y) (9 x² + 12 xy + 16 y²)

Example 3 :

Factorize  m³ + 8

Solution :

m³ + 8  ==>  m³+ 2³

We can compare the above question with the formula a³+ b³. From this we have to expand this using the formula (a + b) (a² - ab + b²).

Instead of "a" we have "m" and instead of "b" we have "2"

=  (m + 2) [m² + (m) (2) + (2)²]

=  (m + 2) (m² + 2 m + 4)

Example 4 :

Factorize  a³ + 125

Solution :

a³ + 125  ==>  a³+ 5³

We can compare the above question with the formula a³+ b³. From this we have to expand this using the formula (a + b) (a² - ab + b²).

Instead of "a" we have "a" and instead of "b" we have "5"

=  (a + 5) [a² + (a) (5) + (5)²]

=  (a + 2) (a² + 5 a + 25)

Example 5 :

Factorize  x³ - 8 y³

Solution :

x³ - 8 y³ = x³ - 2³ y³ ==>  x³+ (2y)³

We can compare the above question with the formula a³+ b³. From this we have to expand this using the formula (a - b) (a² + ab + b²).

Instead of "a" we have "x" and instead of "b" we have "2y"

=  (x + 2y) [x² + (x) (2y) + (2y)²]

=  (x + 2y) (x² + 2 xy + 4 y²)

After having gone through the stuff given above, we hope that the students would have understood "Factor sums and differences of cubes".

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