What is the Factorization of y3+8?
Answers
1.1 Factoring: y3-8
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 : 8 is the cube of 2
Check : y3 is the cube of y1
Factorization is :
(y - 2) • (y2 + 2y + 4)
Trying to factor by splitting the middle term
1.2 Factoring y2 + 2y + 4
The first term is, y2 its coefficient is 1 .
The middle term is, +2y 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 !!
Conclusion : Trinomial can not be factored
Final result :
(y - 2) • (y2 + 2y + 4)
Step-by-step explanation:
=y^3+2^3
=(a^3+b^3)=(a+b)(a^2-ab+b^2) ( identify 5)
=substituting the value
=(y+2)(y^2-y×2+2^2)
=(y+2)(y^2-2y+4)