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Difficulty: MediumOperations on Polynomials

A landscape architect designs a square courtyard with a side length of 4x−34x - 3 meters. A square garden bed with a side length of 2x−12x - 1 meters is placed in one of the corners of the courtyard. The remaining area of the courtyard is paved. If the paved area, in square meters, is represented by the polynomial ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants, what is the value of bb?

Answer: -20

Answer

The value of bb is −20-20.
The remaining paved area is the difference between the area of the courtyard and the garden bed: (16x2−24x+9)−(4x2−4x+1)=12x2−20x+8(16x^2 - 24x + 9) - (4x^2 - 4x + 1) = 12x^2 - 20x + 8. The coefficient of the linear xx term is −20-20.

Step-by-Step Solution

1
Find the polynomial representing the total area of the courtyard.
Atotal=(4x−3)2=16x2−24x+9A_{\text{total}} = (4x - 3)^2 = 16x^2 - 24x + 9 square meters
The area of a square is the square of its side length. We expand (4x−3)2(4x - 3)^2 using the identity (p−q)2=p2−2pq+q2(p - q)^2 = p^2 - 2pq + q^2.
2
Find the polynomial representing the area of the garden bed.
Agarden=(2x−1)2=4x2−4x+1A_{\text{garden}} = (2x - 1)^2 = 4x^2 - 4x + 1 square meters
The garden bed is also a square, so we expand (2x−1)2(2x - 1)^2 using the binomial squaring identity.
3
Subtract the garden bed area from the total area to find the paved area.
Apaved=12x2−20x+8A_{\text{paved}} = 12x^2 - 20x + 8 square meters
The paved area is the difference between the two areas. We distribute the negative sign to all terms of the subtracted polynomial: −(4x2−4x+1)=−4x2+4x−1-(4x^2 - 4x + 1) = -4x^2 + 4x - 1, and then combine like terms.
4
Identify the coefficient of the xx term, bb.
b=−20b = -20
Comparing 12x2−20x+812x^2 - 20x + 8 to ax2+bx+cax^2 + bx + c, the coefficient of the linear xx term is −20-20.

Key Concept

Operations on Polynomials

Alternative Method

To find only the coefficient of the linear term, expand and subtract only the linear terms from both binomial expansions: 2(4x)(−3)−2(2x)(−1)=−24x−(−4x)=−20x2(4x)(-3) - 2(2x)(-1) = -24x - (-4x) = -20x. The coefficient bb is therefore −20-20.
Estimated Time:1m 30s
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Topics

Operations on Polynomials

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