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Difficulty: Very hardOperations on Polynomials

A rectangular region has a width of 2x−32x - 3 meters and a length of 3x+13x + 1 meters. A square piece with a side length of x−2x - 2 meters is removed from the region. The area, in square meters, of the remaining region can be expressed in the standard polynomial form Ax2+Bx+CAx^2 + Bx + C, where AA, BB, and CC are integers. What is the value of the coefficient BB?

Answer: -3

Answer

The coefficient of the linear term, BB, is −3-3.
Subtracting the area of the square, x2−4x+4x^2 - 4x + 4, from the area of the rectangle, 6x2−7x−36x^2 - 7x - 3, yields 5x2−3x−75x^2 - 3x - 7. Thus, the coefficient BB of the xx term is −3-3.

Step-by-Step Solution

1
Calculate the area of the rectangle.
Area = 6x2−7x−36x^2 - 7x - 3
The area of a rectangle is found by multiplying its length and width: (2x−3)(3x+1)=6x2+2x−9x−3=6x2−7x−3(2x - 3)(3x + 1) = 6x^2 + 2x - 9x - 3 = 6x^2 - 7x - 3.
2
Calculate the area of the square.
Area = x2−4x+4x^2 - 4x + 4
The area of a square is the square of its side length: (x−2)2=(x−2)(x−2)=x2−4x+4(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4.
3
Subtract the square's area from the rectangle's area.
Remaining Area = 5x2−3x−75x^2 - 3x - 7
Subtracting the area of the removed square from the total area requires distributing the negative sign to each term of the square's polynomial: (6x2−7x−3)−(x2−4x+4)=6x2−7x−3−x2+4x−4=5x2−3x−7(6x^2 - 7x - 3) - (x^2 - 4x + 4) = 6x^2 - 7x - 3 - x^2 + 4x - 4 = 5x^2 - 3x - 7.
4
Identify the coefficient BB.
B=−3B = -3
In the standard quadratic form Ax2+Bx+CAx^2 + Bx + C, the coefficient of the linear term xx is BB, which corresponds to −3-3 in the polynomial 5x2−3x−75x^2 - 3x - 7.

Key Concept

Operations on Polynomials (multiplication, squaring binomials, and subtraction with negative sign distribution)
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Topics

Operations on Polynomials

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