A large rectangle has a length of 4x + 3 inches and a width of 3x - 2 inches. A smaller re… — ACT

A large rectangle has a length of $4x + 3$ inches and a width of $3x - 2$ inches. A smaller rectangular region is cut out from the center. The cut-out region has a length of $2x - 1$ inches and a width of $x - 5$ inches. The area of the remaining region can be written as the polynomial $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Answer: 12

Answer

The correct answer is 12, which is the coefficient of the linear term in the simplified remaining area polynomial.

The remaining area is found by subtracting the area of the smaller rectangle from the area of the larger rectangle. The area of the larger rectangle is

(4x + 3)(3x - 2) = 12x^2 + x - 6

. The area of the smaller rectangle is

(2x - 1)(x - 5) = 2x^2 - 11x + 5

. Subtracting the two yields

(12x^2 + x - 6) - (2x^2 - 11x + 5) = 10x^2 + 12x - 11

. Matching this to

Ax^2 + Bx + C

, the coefficient

B

of the linear term is

12

Step-by-Step Solution

Find the area of the outer rectangle.

Area =

12x^2 + x - 6

Multiply the length

(4x + 3)

and the width

(3x - 2)

using the distributive property.

Find the area of the inner rectangle.

Area =

2x^2 - 11x + 5

Multiply the length

(2x - 1)

and the width

(x - 5)

using the distributive property.

Subtract the inner area from the outer area to find the remaining area.

Remaining Area =

10x^2 + 12x - 11

Subtract the polynomial

(2x^2 - 11x + 5)

from

(12x^2 + x - 6)

by distributing the negative sign and combining like terms.

Identify the coefficient

B

of the

x

term.

B = 12

Match the simplified polynomial

10x^2 + 12x - 11

with the form

Ax^2 + Bx + C

Key Concept

Operations on Polynomials (multiplication and subtraction of polynomials)

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