prove that (8a^{6}) + 9a^{3} + 8 = 0
Answers
Answer:
Changes made to your input should not affect the solution:
(1): "a3" was replaced by "a^3". 1 more similar replacement(s).
STEP
1
:
Equation at the end of step 1
((a6) - 32a3) + 8
STEP
2
:
Trying to factor by splitting the middle term
2.1 Factoring a6-9a3+8
The first term is, a6 its coefficient is 1 .
The middle term is, -9a3 its coefficient is -9 .
The last term, "the constant", is +8
Step-1 : Multiply the coefficient of the first term by the constant 1 • 8 = 8
Step-2 : Find two factors of 8 whose sum equals the coefficient of the middle term, which is -9 .
-8 + -1 = -9 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and -1
a6 - 8a3 - 1a3 - 8
Step-4 : Add up the first 2 terms, pulling out like factors :
a3 • (a3-8)
Add up the last 2 terms, pulling out common factors :
1 • (a3-8)
Step-5 : Add up the four terms of step 4 :
(a3-1) • (a3-8)
Which is the desired factorization
Trying to factor as a Difference of Cubes:
2.2 Factoring: a3-1
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 1 is the cube of 1
Check : a3 is the cube of a1
Factorization is :
(a - 1) • (a2 + a + 1)
Trying to factor by splitting the middle term
2.3 Factoring a2 + a + 1
The first term is, a2 its coefficient is 1 .
The middle term is, +a its coefficient is 1 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is 1 .
-1 + -1 = -2
1 + 1 = 2
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Trying to factor as a Difference of Cubes:
2.4 Factoring: a3-8
Check : 8 is the cube of 2
Check : a3 is the cube of a1
Factorization is :
(a - 2) • (a2 + 2a + 4)
Trying to factor by splitting the middle term
2.5 Factoring a2 + 2a + 4
The first term is, a2 its coefficient is 1 .
The middle term is, +2a its coefficient is 2 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is 2 .
-4 + -1 = -5
-2 + -2 = -4
-1 + -4 = -5
1 + 4 = 5
2 + 2 = 4
4 + 1 = 5
Observation : No two such factors can be found !!