Essentially two polynomials are equivalent if they yield the same output for every input. We can create equivalent expressions by performing logical operations.

We are given the expression `-4x^3+x^2-6x+8` and we are asked to find, from a potential list of equivalent expressions, the expression that is not equivalent to the...

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Essentially two polynomials are equivalent if they yield the same output for every input. We can create equivalent expressions by performing logical operations.

We are given the expression `-4x^3+x^2-6x+8` and we are asked to find, from a potential list of equivalent expressions, the expression that is not equivalent to the given expression.

(1) Suppose we factor out a common `x^2` from the first two terms and a common -2 from the last two terms. Factoring out in this manner is essentially applying the distributive property.

Then we get `x^2(-4x+1)` and `-2(3x-4)` . Since this is an allowed logical operation `x^2(-4x+1)-2(3x-4) -= -4x^3+x^2-6x+8` **(The expressions are equivalent.)**

(2) Given `x(-4x^2-x+6)+8` we can use the distributive property to obtain `x(-4x^2-x+6)+8 -= -4x^3-x^2+6x+8` . Note that the signs on the quadratic term (x^2) and the linear term (6x) are not the same as the given polynomial expression. Thus **this expression is not equivalent to the given expression.**

(3) Suppose we factor the trinomial formed by the last three terms:

`-4x^3+[x^2-6x+8] -= -4x^3+[(x-4)(x-2)]` **The two expressions are equivalent.**

(4) Suppose we rearrange the terms of the given polynomial. This is allowed since addition is commutative.

`-4x^3+x^2-6x+8 -= -4x^3+8+x^2-6x`

Now factor out the common factor of -4 in the first two terms and the common factor of x in the last two terms to get:

`-=-4(x^3-2)+x(x-6)`

Since equivalence is transitive **the expressions are equivalent.**

**Of the four choices, choice two is not equivalent to the given polynomial.**