Sure, let's factorize each expression step-by-step:
### (i) Factorise [tex]\( 27y^3 + 125z^3 \)[/tex]
1. Identify as a Sum of Cubes:
Recognize that both terms are perfect cubes:
[tex]\( 27y^3 = (3y)^3 \)[/tex]
and
[tex]\( 125z^3 = (5z)^3 \)[/tex].
2. Apply the Sum of Cubes Formula:
The sum of cubes can be factored using the formula:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
Here, [tex]\( a = 3y \)[/tex] and [tex]\( b = 5z \)[/tex]. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
(3y)^3 + (5z)^3 = (3y + 5z)\left((3y)^2 - (3y)(5z) + (5z)^2\right)
\][/tex]
3. Simplify Inside the Parentheses:
[tex]\[
(3y + 5z)\left(9y^2 - 15yz + 25z^2\right)
\][/tex]
4. Final Factored Form:
[tex]\[
27y^3 + 125z^3 = (3y + 5z)(9y^2 - 15yz + 25z^2)
\][/tex]
### (ii) Factorise [tex]\( 27x^3 + y^3 + z^3 - 9xyz \)[/tex]
1. Recognize Special Form:
The expression can be recognized as a form of a generalized sum of cubes combined with a particular product term, suggestive of the factorization formula for three variables:
[tex]\[
a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
\][/tex]
where [tex]\( a = 3x \)[/tex], [tex]\( b = y \)[/tex], and [tex]\( c = z \)[/tex].
2. Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\(
27x^3 + y^3 + z^3 - 9xyz = (3x + y + z)\left((3x)^2 + y^2 + z^2 - (3x)y - yz - (3x)z\right)
\)[/tex]
3. Simplify Inside the Parentheses:
[tex]\[
(3x + y + z)\left(9x^2 + y^2 + z^2 - 3xy - yz - 3xz\right)
\][/tex]
4. Final Factored Form:
[tex]\[
27x^3 + y^3 + z^3 - 9xyz = (3x + y + z)(9x^2 - 3xy - 3xz + y^2 - yz + z^2)
\][/tex]
Therefore, the factorized forms are:
1. [tex]\( 27y^3 + 125z^3 = (3y + 5z)(9y^2 - 15yz + 25z^2) \)[/tex]
2. [tex]\( 27x^3 + y^3 + z^3 - 9xyz = (3x + y + z)(9x^2 - 3xy - 3xz + y^2 - yz + z^2) \)[/tex]
