When the expression 3x^2(2x - y) - 2y(x^2 - 3xy) - (4x^3 - xy^2) is simplified to the form… — ACT

When the expression $3x^2(2x - y) - 2y(x^2 - 3xy) - (4x^3 - xy^2)$ is simplified to the form $Ax^3 + Bx^2y + Cxy^2$ , where $A$ , $B$ , and $C$ are integers, what is the value of $A + B + C$ ?

A
-8
B
-2
C
1
D
2
4Answer

Answer

The correct sum of the coefficients is 4. By distributing all terms correctly:

3x^2(2x - y) = 6x^3 - 3x^2y

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

, and

-(4x^3 - xy^2) = -4x^3 + xy^2

. Combining the terms yields

2x^3 - 5x^2y + 7xy^2

, which gives

A = 2

B = -5

, and

C = 7

. Summing these values gives

2 - 5 + 7 = 4

Step-by-Step Solution

Distribute

3x^2

to the first parenthetical expression:

3x^2(2x - y)

6x^3 - 3x^2y

Applying the distributive property multiplies

3x^2

by both

2x

and

-y

Distribute

-2y

to the second parenthetical expression:

-2y(x^2 - 3xy)

-2x^2y + 6xy^2

Applying the distributive property multiplies

-2y

by both

x^2

and

-3xy

, noting that a negative times a negative is a positive.

Distribute the negative sign to the third parenthetical expression:

-(4x^3 - xy^2)

-4x^3 + xy^2

Distributing the negative sign changes the signs of both terms inside the parenthesis.

Combine the expanded parts:

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

and group like terms

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

Grouping like terms together makes it easier to combine their coefficients.

Combine the coefficients of the like terms

2x^3 - 5x^2y + 7xy^2

Combining the coefficients gives

A = 2

B = -5

, and

C = 7

Calculate the sum

A + B + C

2 + (-5) + 7 = 4

Adding the coefficients together yields the final numerical value.

Key Concept

Simplifying algebraic expressions by distributing coefficients (including negative signs) and combining like terms.

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