Which expression is not equivalent to -4x^3 + x^2 - 6x + 8? (1) x^2 (-4x+1)-2(3x-4) (2) x(-4x^2 - x + 6) + 8 (3) -4x^3 + (x - 2)(x - 4) (4) -4(x^3 - 2) + x(x - 6)

Quick answer:

(2) x(-4x^2-x+6)+8 is not equivalent to the expression -4x^3+x^2-6x+8

Expert Answers

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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.

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