Answer :
To factorize the expression [tex]\( a^3 + 2\sqrt{2} b^3 \)[/tex], we need to express it in a product of polynomials if possible.
### Step-by-Step Solution:
1. Identify the Structure:
The given expression is [tex]\( a^3 + 2\sqrt{2} b^3 \)[/tex]. Notice that it resembles a sum of cubes, but with a slight variation due to the coefficient [tex]\( 2\sqrt{2} \)[/tex].
2. Recall the Factorization Formula for Sum of Cubes:
The standard factorization for a sum of cubes [tex]\( x^3 + y^3 \)[/tex] is:
[tex]\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \][/tex]
We need to apply this to our expression by making it fit this form. In this case, our expression can be seen in the form [tex]\( a^3 + (2\sqrt{2}b)^3 \)[/tex].
3. Rewrite the Expression in Sum of Cubes Format:
We have:
[tex]\[ a^3 + (2\sqrt{2} b)^3 \][/tex]
Here, [tex]\( x = a \)[/tex] and [tex]\( y = 2\sqrt{2} b \)[/tex].
4. Apply the Sum of Cubes Formula:
Using the sum of cubes formula, substitute [tex]\( x = a \)[/tex] and [tex]\( y = 2\sqrt{2} b \)[/tex]:
[tex]\[ a^3 + (2\sqrt{2} b)^3 = (a + 2\sqrt{2} b)\left[ a^2 - a(2\sqrt{2} b) + (2\sqrt{2} b)^2 \right] \][/tex]
5. Simplify the Expression Inside the Brackets:
[tex]\[ = (a + 2\sqrt{2} b) \left[ a^2 - 2\sqrt{2} ab + (2\sqrt{2} b)^2 \right] \][/tex]
Calculate [tex]\( (2\sqrt{2} b)^2 \)[/tex]:
[tex]\[ (2\sqrt{2} b)^2 = 4 \times 2 \times b^2 = 8 b^2 \][/tex]
Thus, the expression inside the brackets becomes:
[tex]\[ a^2 - 2\sqrt{2} ab + 8b^2 \][/tex]
6. Combine Results:
Therefore, the factorized form is:
[tex]\[ a^3 + 2\sqrt{2} b^3 = (a + 2\sqrt{2} b) (a^2 - 2\sqrt{2} ab + 8 b^2) \][/tex]
### Conclusion:
The factorized form of the expression [tex]\( a^3 + 2\sqrt{2} b^3 \)[/tex] is:
[tex]\[ a^3 + 2 \sqrt{2} b^3 = (a + 2\sqrt{2} b) (a^2 - 2\sqrt{2} ab + 8 b^2) \][/tex]
### Step-by-Step Solution:
1. Identify the Structure:
The given expression is [tex]\( a^3 + 2\sqrt{2} b^3 \)[/tex]. Notice that it resembles a sum of cubes, but with a slight variation due to the coefficient [tex]\( 2\sqrt{2} \)[/tex].
2. Recall the Factorization Formula for Sum of Cubes:
The standard factorization for a sum of cubes [tex]\( x^3 + y^3 \)[/tex] is:
[tex]\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \][/tex]
We need to apply this to our expression by making it fit this form. In this case, our expression can be seen in the form [tex]\( a^3 + (2\sqrt{2}b)^3 \)[/tex].
3. Rewrite the Expression in Sum of Cubes Format:
We have:
[tex]\[ a^3 + (2\sqrt{2} b)^3 \][/tex]
Here, [tex]\( x = a \)[/tex] and [tex]\( y = 2\sqrt{2} b \)[/tex].
4. Apply the Sum of Cubes Formula:
Using the sum of cubes formula, substitute [tex]\( x = a \)[/tex] and [tex]\( y = 2\sqrt{2} b \)[/tex]:
[tex]\[ a^3 + (2\sqrt{2} b)^3 = (a + 2\sqrt{2} b)\left[ a^2 - a(2\sqrt{2} b) + (2\sqrt{2} b)^2 \right] \][/tex]
5. Simplify the Expression Inside the Brackets:
[tex]\[ = (a + 2\sqrt{2} b) \left[ a^2 - 2\sqrt{2} ab + (2\sqrt{2} b)^2 \right] \][/tex]
Calculate [tex]\( (2\sqrt{2} b)^2 \)[/tex]:
[tex]\[ (2\sqrt{2} b)^2 = 4 \times 2 \times b^2 = 8 b^2 \][/tex]
Thus, the expression inside the brackets becomes:
[tex]\[ a^2 - 2\sqrt{2} ab + 8b^2 \][/tex]
6. Combine Results:
Therefore, the factorized form is:
[tex]\[ a^3 + 2\sqrt{2} b^3 = (a + 2\sqrt{2} b) (a^2 - 2\sqrt{2} ab + 8 b^2) \][/tex]
### Conclusion:
The factorized form of the expression [tex]\( a^3 + 2\sqrt{2} b^3 \)[/tex] is:
[tex]\[ a^3 + 2 \sqrt{2} b^3 = (a + 2\sqrt{2} b) (a^2 - 2\sqrt{2} ab + 8 b^2) \][/tex]
