3y3+192=? Answer-: STEP1:Equation at the end of step 1 3y3 –

3y3+192=?

Answer-:
STEP
1
:
Equation at the end of step 1
3y3 – 192
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :

3y3 – 192 = 3 • (y3 – 64)

Trying to factor as a Difference of Cubes:
3.2 Factoring: y3 – 64

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 : 64 is the cube of 4
Check : y3 is the cube of y1

Factorization is :
(y – 4) • (y2 + 4y + 16)

Trying to factor by splitting the middle term
3.3 Factoring y2 + 4y + 16

The first term is, y2 its coefficient is 1 .
The middle term is, +4y its coefficient is 4 .
The last term, “the constant”, is +16

Step-1 : Multiply the coefficient of the first term by the constant 1 • 16 = 16

Step-2 : Find two factors of 16 whose sum equals the coefficient of the middle term, which is 4 .

-16 + -1 = -17
-8 + -2 = -10
-4 + -4 = -8
-2 + -8 = -10
-1 + -16 = -17
1 + 16 = 17
2 + 8 = 10
4 + 4 = 8
8 + 2 = 10
16 + 1 = 17

Observation : No two such factors can be found
Conclusion : Trinomial can not be factored

Final result :
3 • (y – 4) • (y2 + 4y + 16)​


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