Polynomials (add, multiply, factor) Transitional Reasoning Download PDF

Questions

1. 1.

   A multimedia design studio installs a custom video wall constructed from two identical square LED panels and three identical rectangular LED panels. For a design parameter $x > 1$, each square panel has a side length of $x + 2$ decimeters, and each rectangular panel has a width of $x + 2$ decimeters and a length of $2x + 5$ decimeters. Which expression represents the completely factored polynomial for the total active display area of the video wall in square decimeters?

   • A) $(x + 2)(8x + 19)$
   • B) $(x + 2)(7x + 17)$
   • C) $(x + 2)(4x + 9)$
   • D) $(x + 2)(8x + 9)$
2. 2.

   An environmental engineering team designs a multi-stage bio-filtration wetland composed of three sequential treatment zones. For an operational scaling parameter $x > 1$, the surface areas (in square meters) of the three zones are modeled by $A_1(x) = 3x^2 + 8x + 4$, $A_2(x) = 4x^2 + 9x + 5$, and $A_3(x) = x^2 + 5x + 6$. If the total surface area of the wetland is the sum of these three zones and forms a single rectangular basin with linear dimensions having integer coefficients, which expression represents the completely factored form of the total surface area in square meters?

   • A) $(x + 3)(8x + 5)$
   • B) $(2x + 3)(4x + 5)$
   • C) $(2x + 5)(4x + 3)$
   • D) $(x + 1)(8x + 15)$
3. 3.

   An industrial packaging facility manufactures custom storage crates whose interior base area is modeled by the polynomial $A(x) = 2x^2 + 5x + 3$ square decimeters and whose height is $h(x) = 3x - 2$ decimeters for $x > 2$. To protect fragile equipment during transit, a molded foam insert with a volume of $V_{\text{insert}}(x) = 4x^2 + 7x - 6$ cubic decimeters is secured inside each crate. Which polynomial expression in standard form represents the remaining usable interior volume of the crate in cubic decimeters?

   • A) $6x^3 + 7x^2 - 8x - 12$
   • B) $6x^3 + 15x^2 + 6x - 12$
   • C) $6x^3 + 7x^2 - 8x$
   • D) $6x^3 + 7x^2 + 12x$
4. 4.

   A logistics company manufactures an open-top rectangular storage bin whose base area is modeled by $B(x) = x^2 + 9x + 14$ square decimeters. The bin has a height of $2x + 3$ decimeters. To organize inventory, a flat rectangular dividing wall of the same height is installed across the narrower dimension of the base. For all $x > 0$, which polynomial expression in standard form represents the area of this dividing wall in square decimeters?

   • A) $2x^2 + 7x + 6$
   • B) $2x^2 + 17x + 21$
   • C) $2x^2 + 11x + 12$
   • D) $2x^2 + 5x + 3$
5. 5.

   An industrial heating system generates thermal energy modeled by $E_{\text{gen}}(x) = (2x + 3)(3x + 7)$ megajoules for a production batch of scale $x$. During operation, internal cooling subsystems absorb an amount of energy modeled by $E_{\text{loss}}(x) = 3x^2 + 4x + 1$ megajoules. If the net available thermal energy is given by $E_{\text{net}}(x) = E_{\text{gen}}(x) - E_{\text{loss}}(x)$, which expression represents the completely factored form of $E_{\text{net}}(x)$?

   • A) $(3x + 5)(x + 4)$
   • B) $(3x + 4)(x + 5)$
   • C) $(3x + 2)(x + 10)$
   • D) $(3x + 10)(x + 2)$
6. 6.

   An engineering firm designs a hollow structural beam whose rectangular cross-section has outer dimensions of $3x + 4$ centimeters by $2x + 5$ centimeters. A hollow rectangular core with dimensions of $x + 2$ centimeters by $2x + 1$ centimeters runs through the center of the beam. Which expression represents the completely factored cross-sectional area of the solid material surrounding the hollow core in square centimeters?

   • A) $2(2x + 9)(x + 1)$
   • B) $2(4x + 3)(x + 3)$
   • C) $2(2x - 3)(x - 3)$
   • D) $2(2x + 3)(x + 3)$
7. 7.

   A community center combines two adjacent rectangular courtyards that share a common boundary of length $x + 3$ meters. The area of the first courtyard is $A_1(x) = x^2 + 8x + 15$ square meters, and the area of the second courtyard is $A_2(x) = 2x^2 + 7x + 3$ square meters. If the courtyards form a single unified rectangular space along this shared boundary, which expression represents the total length of the adjacent side?

   • A) $3x + 4$
   • B) $3x + 8$
   • C) $2x + 6$
   • D) $3x + 6$