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Difficulty: MediumOperations on Polynomials

A computer hardware engineer is designing a rectangular silicon microchip. The total surface area of the microchip, in square millimeters, is represented by the expression 2y(3y2−4y+5)2y(3y^2 - 4y + 5). A square sensor is embedded in the center of the chip, occupying an area represented by the expression (2y−3)2(2y - 3)^2 square millimeters. Which of the following expressions represents the remaining surface area of the microchip, in square millimeters, that is not occupied by the sensor?

  1. A
    6y3−12y2+10y−96y^3 - 12y^2 + 10y - 9
  2. B
    6y3−12y2−2y+96y^3 - 12y^2 - 2y + 9
  3. 6y3−12y2+22y−96y^3 - 12y^2 + 22y - 9Answer
  4. D
    −6y2+22y−9-6y^2 + 22y - 9
  5. E
    6y3−12y2+22y+96y^3 - 12y^2 + 22y + 9

Answer

The correct expression is 6y3−12y2+22y−96y^3 - 12y^2 + 22y - 9.
To find the remaining surface area of the microchip, subtract the area of the sensor from the total area. First, expand the expression for the total area: 2y(3y2−4y+5)=6y3−8y2+10y2y(3y^2 - 4y + 5) = 6y^3 - 8y^2 + 10y. Next, expand the expression for the sensor area: (2y−3)2=4y2−12y+9(2y - 3)^2 = 4y^2 - 12y + 9. Subtract the sensor area from the total area, distributing the negative sign to all terms: (6y3−8y2+10y)−(4y2−12y+9)=6y3−8y2+10y−4y2+12y−9(6y^3 - 8y^2 + 10y) - (4y^2 - 12y + 9) = 6y^3 - 8y^2 + 10y - 4y^2 + 12y - 9. Combining like terms yields the expression 6y3−12y2+22y−96y^3 - 12y^2 + 22y - 9.

Step-by-Step Solution

1
Expand the total area of the microchip by distributing the monomial 2y2y to each term of the trinomial 3y2−4y+53y^2 - 4y + 5.
2y(3y2−4y+5)=6y3−8y2+10y2y(3y^2 - 4y + 5) = 6y^3 - 8y^2 + 10y
This determines the polynomial representing the total area before any components are subtracted.
2
Expand the area of the square sensor by squaring the binomial (2y−3)(2y - 3).
(2y−3)2=4y2−12y+9(2y - 3)^2 = 4y^2 - 12y + 9
This determines the polynomial representing the area occupied by the sensor, ensuring the middle term −12y-12y is correctly calculated.
3
Subtract the sensor's area from the total area, ensuring the negative sign is distributed to every term in the sensor's polynomial.
(6y3−8y2+10y)−(4y2−12y+9)=6y3−8y2+10y−4y2+12y−9(6y^3 - 8y^2 + 10y) - (4y^2 - 12y + 9) = 6y^3 - 8y^2 + 10y - 4y^2 + 12y - 9
This sets up the subtraction of the two polynomials while distributing the negative sign to avoid common sign errors.
4
Combine like terms to find the final simplified expression in standard descending order.
6y3−12y2+22y−96y^3 - 12y^2 + 22y - 9
This combines the cubic, quadratic, linear, and constant terms to find the remaining area.

Key Concept

Polynomial operations including monomial multiplication, binomial expansion, and subtraction with sign distribution.
Estimated Time:1m 30s
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Topics

Operations on Polynomials

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