When the expression 2x(x^2 - 3xy + 2y^2) - (x - 2y)^3 - y(x^2 - 5xy + 4y^2) is fully simpl… — ACT

When the expression $2x(x^2 - 3xy + 2y^2) - (x - 2y)^3 - y(x^2 - 5xy + 4y^2)$ is fully simplified by combining like terms, what is the coefficient of the $xy^2$ term?

Answer: -3

Answer

The coefficient of the

xy^2

term is

-3

Expanding the terms of the expression yields

2x^3 - 6x^2y + 4xy^2

-x^3 + 6x^2y - 12xy^2 + 8y^3

, and

-x^2y + 5xy^2 - 4y^3

. Summing the coefficients of the

xy^2

terms gives

4 - 12 + 5 = -3

. Thus, the coefficient of

xy^2

-3

Step-by-Step Solution

Expand the first term of the expression:

2x(x^2 - 3xy + 2y^2)

2x^3 - 6x^2y + 4xy^2

Distribute the monomial

2x

to each term inside the parentheses:

2x \cdot x^2 = 2x^3

2x \cdot (-3xy) = -6x^2y

, and

2x \cdot 2y^2 = 4xy^2

Expand the cubed binomial

(x - 2y)^3

and distribute the negative sign

-x^3 + 6x^2y - 12xy^2 + 8y^3

First, expand the binomial

(x - 2y)^3 = x^3 - 3(x^2)(2y) + 3(x)(2y)^2 - (2y)^3 = x^3 - 6x^2y + 12xy^2 - 8y^3

. Then, distribute the negative sign to all terms inside the parentheses.

Expand the third term:

-y(x^2 - 5xy + 4y^2)

-x^2y + 5xy^2 - 4y^3

Distribute the negative monomial

-y

to each term inside the parentheses:

-y \cdot x^2 = -x^2y

-y \cdot (-5xy) = 5xy^2

, and

-y \cdot 4y^2 = -4y^3

Group and combine all like terms

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

Combine the coefficients of the corresponding terms:

(2x^3 - x^3) + (-6x^2y + 6x^2y - x^2y) + (4xy^2 - 12xy^2 + 5xy^2) + (8y^3 - 4y^3) = x^3 - x^2y - 3xy^2 + 4y^3

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:3m 0s

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