factorise 9- a^6+2a^3b^3-b^6
To factorize the given expression 9 - a^6 + 2a^3b^3 - b^6, we can use the difference of cubes formula and grouping.
First, let's arrange the terms as a difference of cubes:
9 - a^6 + 2a^3b^3 - b^6
= (3)^2 - (a^2)^3 + 2(a^3b^3) - (b^2)^3
Now, we can apply the difference of cubes formula, which states that a^3 - b^3 = (a - b)(a^2 + ab + b^2):
= [(3) - (a^2)][(3)^2 + (3)(a^2) + (a^2)^2] + 2(a^3b^3 - b^6)
Simplifying the expression, we get:
= (3 - a^2)(9 + 3a^2 + a^4) + 2(a^3b^3 - b^6)
Now, let's group the terms:
= (3 - a^2)[(9 + 3a^2 + a^4) + 2(a^3b^3 - b^6)]
We can see that we have a common factor of (3 - a^2), so we can factor it out:
= (3 - a^2)(9 + 3a^2 + a^4 + 2a^3b^3 - 2b^6)
Therefore, the factorized form of the expression 9 - a^6 + 2a^3b^3 - b^6 is (3 - a^2)(9 + 3a^2 + a^4 + 2a^3b^3 - 2b^6).
To factorize the given expression 9 - a^6 + 2a^3b^3 - b^6, we can use the difference of cubes factorization formula.
The difference of cubes formula states that for any two terms a^3 and b^3, we can factorize the expression a^3 - b^3 as (a - b)(a^2 + ab + b^2).
Now, let's apply the difference of cubes formula to factorize the given expression:
9 - a^6 + 2a^3b^3 - b^6
= (3)^2 - (a^2)^3 + 2(a^3b^3) - (b^2)^3
Using the difference of cubes formula, we can write the expression as:
= (3 - a^2)(9 + 3a^2 + a^4) + 2(a^3b^3 - b^6)
Now, we can factorize the remaining terms a^3b^3 - b^6 using the difference of cubes formula:
= (3 - a^2)(9 + 3a^2 + a^4) + 2(b^3 - a^3)(b^3 + a^3)
Finally, the fully factorized expression becomes:
= (3 - a^2)(9 + 3a^2 + a^4) + 2(b - a)(b^2 + ab + a^2)(b^3 + a^3)
So, the factorized form of the expression 9 - a^6 + 2a^3b^3 - b^6 is:
(3 - a^2)(9 + 3a^2 + a^4) + 2(b - a)(b^2 + ab + a^2)(b^3 + a^3)
9 - a⁶ + 2 a³ b³ - b⁶ =
- a⁶ + 2 a³ b³ - b⁶ + 9 =
- ( a⁶ - 2 a³ b³ + b⁶ - 9 )
____________
Substitution:
u = a³
v = b³
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- ( a⁶ - 2 a³ b³ + b⁶ - 9 ) =
- ( u² - 2 u v + v² - 9 ) =
_____________________
Remark:
u² - 2 u v + v² = ( u - v )²
_____________________
- [ ( u - v )² - 9 ] =
- [ ( u - v )² - 3² ]
_______________
New substitution:
z = u - v
w = 3
_______________
- [ ( u - v )² - 3² ] =
- ( z² - w² ) =
_________________________
Remark:
( z² - w² ) = ( z + w ) ( z - w )
_________________________
- ( z + w ) ( z - w ) =
- ( u - v + 3 ) ( u - v - 3 )
Replace u = a³ and v = b³ in this exspression:
- ( u - v + 3 ) ( u - v - 3 ) =
- ( a³ - b³ + 3 ) ( a³ - b³ - 3 )
So:
9 - a⁶ + 2 a³ b³ - b⁶ = - ( a³ - b³ + 3 ) ( a³ - b³ - 3 )
9-a^6+2a^3b^3-b^6
= 9 - (a^6-2a^3b^3+b^6)
= 9 - (a^3-b^3)^2
= (3-(a^3-b^3))(3+(a^3-b^3))