3x3 − 4x − 2x3 − x can be put in the form Ax + Bx − 1 + Cx + 1,where A, B and C are constants.TrueFalse

To determine if the expression $3x^3 - 4x - 2x^3 - x$ can be put in the form $Ax + Bx^{-1} + Cx + 1$, let's first simplify the expression and analyze the components.

Step 1: Combine Like Terms

Starting with the original expression:

$3x^3 - 4x - 2x^3 - x$

Combine the $x^3$ terms and the $x$ terms:

$(3x^3 - 2x^3) + (-4x - x) = x^3 - 5x$

Step 2: Analyze the Required Form

The form we want is:

$Ax + Bx^{-1} + Cx + 1$

This includes:

• Terms with positive powers of $x$ (like $Ax$ and $Cx$),
• A term with negative power of $x$ ($Bx^{-1}$),
• A constant term (1).

Step 3: Identify Components

The simplified expression $x^3 - 5x$ only includes a polynomial with positive powers of $x$ (specifically $x^3$ and $x^1$). There are no terms like $Bx^{-1}$ (with negative power) or a standalone constant.

Conclusion

Since the expression cannot include the required terms to match the format $Ax + Bx^{-1} + Cx + 1$ due to the absence of $Bx^{-1}$ and the presence of the $x^3$ term, we conclude that it cannot be expressed in that form.

Final Answer

The statement is False.