State that identity is true: 3+(4+x)^3=x^3+12x^2+115

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justaguide eNotes educator| Certified Educator

We have to prove the identity 3+(4+x)^3=x^3+12x^2+115

Starting with the left hand side

3+(4+x)^3

=> 3 + 4^3 + x^3 + 3*16*x + 3*4*x^2

=> 3 + 64 + x^3 + 48x + 12x^2

=> x^3 + 12x^2 + 48x + 67

We see that this is not equal to the right hand side.

Therefore the expression that we have is not an identity that can be proved.

giorgiana1976 | Student

We'll manage the left side and we'll expand the cube:

(4+x)^3 = 4^3 + x^3 + 3*4*x(4+x)

(4+x)^3 = 64 + x^3 + 12*x(4+x)

We'll remove the brackets from the right:

(4+x)^3 = 64 + x^3 + 12*x^2 + 48x

We'll add 3 both sides:

(4+x)^3 + 3 = 64 + x^3 + 12*x^2 + 48x + 3

We'll combine like terms:

(4+x)^3 + 3 = x^3 + 12*x^2 + 48x + 67

We notice that the right side of the identity to be demonstrated does not have the term in x, so, the both sides expressions are  not equivalent.

The given expression does not represent an identity.

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