Polynomials

Exercise 2.5 Part 10

Question: 9 – Verify:

(i) `x^3 + y^3 = (x + y)(x^2 - xy + y^2)`

Answer: RHS = `(x + y)(x^2 - xy + y^2`

`= x^3 - x^2\y + xy^2 + yx^2 - xy^2 + y^3`

`= x^3 + y^3` = LHS proved

(ii) `x^3 + y^3 = (x - y)(x^2 + xy + y^2)`

Answer: RHS `= (x - y)(x^2 + xy + y^2`

`= x^3 + x^2\y + xy^2 - yx^2 - xy^2 - y^3`

`= x^3 - y^3` = LHS Proved

Question: 10 – Factorise each of the following:

(i) `27y^3 + 125z^3`

Answer: Given; `27y^3 + 125z^3`

`= (3y)^3 + (5z)^3`

Using the identity `x^3 + y^3 = (x + y)(x^2 - xy + y^2)`

We get: `27y^3 + 125z^3`

`= (3y + 5z)[(3y)^2 - 3y\xx5z + (5z)^2]`

`= (3y + 5z)(9y^2 - 15yz + 25z^2)`

(ii) `64m^3 - 343n^3`

Answer: Given; `64m^3 - 343n^3`

`= (4m – 7n)[(4m)^2 + 4m\xx7n + (7n)^2]`

`= (4m – 7n)(16m^2 + 28mn + 49n^2)`

Question: 11 – Factorise:

`27x^3 + y^3 + z^3 - 9xy\z`

Answer: Given; `27^3 + y^3 + z^3 - 9xy\z`

`= (3x)^3 + y^3 + z^3 - 3xx3xy\z`

Using the identity `x^3 + y^3 + z^3 - 3xy\z`

`= (x + y + z)(x^2 + y^2 + z^2 - xy – yz – xz)`

We get: `(3x + y + z)[(3x)^2 + y^2 + z^2 - 3xy – yz – 3xz]`

`= (3x + y + z)(9x^2 + y^2 + z^3 - 3xy – yz – 3xz)`