To factor the expression [tex]\( 125 x^3 + 216 y^3 \)[/tex], we'll follow these steps:
1. Identify the form of the expression:
The expression [tex]\( 125 x^3 + 216 y^3 \)[/tex] is a sum of cubes. Recall that the sum of cubes can be factored using the formula:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
2. Rewrite the expression in the form [tex]\( a^3 + b^3 \)[/tex]:
Notice that:
[tex]\[
125 x^3 = (5x)^3 \quad \text{and} \quad 216 y^3 = (6y)^3
\][/tex]
Thus, the expression becomes:
[tex]\[
(5x)^3 + (6y)^3
\][/tex]
3. Apply the sum of cubes formula:
According to our formula:
[tex]\[
(5x)^3 + (6y)^3 = (5x + 6y) \left( (5x)^2 - (5x)(6y) + (6y)^2 \right)
\][/tex]
4. Simplify the factors inside the parentheses:
- Calculate [tex]\((5x)^2\)[/tex]:
[tex]\[
(5x)^2 = 25x^2
\][/tex]
- Calculate [tex]\(-(5x)(6y)\)[/tex]:
[tex]\[
-(5x)(6y) = -30xy
\][/tex]
- Calculate [tex]\((6y)^2\)[/tex]:
[tex]\[
(6y)^2 = 36y^2
\][/tex]
Putting it all together, we have:
[tex]\[
(5x + 6y)(25x^2 - 30xy + 36y^2)
\][/tex]
So, the factorization of the expression [tex]\( 125 x^3 + 216 y^3 \)[/tex] is:
[tex]\[
(5 x + 6 y)( 25 x^{\wedge} 2 - 30 x y + 36 y^{\wedge} 2)
\][/tex]
