Which of the following is equivalent to the expression 4x^2(x - 2y) - 3x(x^2 - 3xy) - (x^3… — ACT

Which of the following is equivalent to the expression $4x^2(x - 2y) - 3x(x^2 - 3xy) - (x^3 + xy^2)$ for all real values of $x$ and $y$ ?

The correct simplified expression is

x^2y - xy^2

Distributing the terms yields

4x^3 - 8x^2y - 3x^3 + 9x^2y - x^3 - xy^2

. Grouping and combining like terms results in

(4-3-1)x^3 + (-8+9)x^2y - xy^2 = x^2y - xy^2

Distribute the term

4x^2

to the terms inside the first set of parentheses:

4x^2(x - 2y)

4x^3 - 8x^2y

Multiplying a monomial by a binomial requires distributing the multiplier to each term inside.

Distribute the term

-3x

to the terms inside the second set of parentheses:

-3x(x^2 - 3xy)

-3x^3 + 9x^2y

Carefully apply the distributive property, remembering that multiplying a negative term by a negative term yields a positive term:

-3x \cdot (-3xy) = 9x^2y

Distribute the negative sign (or

-1

) to the terms inside the third set of parentheses:

-(x^3 + xy^2)

-x^3 - xy^2

Distributing the negative sign changes the signs of all terms inside the parentheses.

Combine all parts and group the like terms:

x^3

x^2y

, and

xy^2

x^2y - xy^2

Add the coefficients of the terms with the same variable parts:

(4 - 3 - 1)x^3 + (-8 + 9)x^2y - xy^2 = 0x^3 + x^2y - xy^2 = x^2y - xy^2

Simplifying algebraic expressions by distributing and combining like terms.

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