Polynomials (add, multiply, factor) Transitional Reasoning Practice

Questions

1. 1.

   An industrial design team manufactures two models of rectangular storage bins that have identical base areas, where $x > 1$. The volume of Model 1 is given by $V_1(x) = 2x^3 + 9x^2 + 7x - 6$ cubic decimeters, and the volume of Model 2 is given by $V_2(x) = 2x^3 + 13x^2 + 17x - 12$ cubic decimeters. If the height of each model is represented by a linear polynomial with integer coefficients, which expression represents the shared base area of the bins, in square decimeters, in completely factored form?

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

   A commercial shipping firm calculates the total freight tonnage transported by two dedicated cargo vessels during an operational cycle, where the fleet parameter satisfies $x > 2$. Vessel Alpha transports $x + 5$ containers, each having an average cargo mass modeled by $x^2 + 2x - 8$ metric tons. Vessel Beta transports $x - 2$ containers, each having an average cargo mass modeled by $2x^2 + 13x + 15$ metric tons. Which expression represents the total freight tonnage transported by both vessels, in metric tons, in completely factored form?

   • A) $(x + 2)(x - 5)(3x + 7)$
   • B) $(x - 2)(x + 5)(3x + 1)$
   • C) $(x - 2)(x + 5)(3x + 7)$
   • D) $(x - 2)(x + 5)(3x + 11)$
3. 3.

   A cloud-computing server cluster executes three concurrent algorithmic workloads during a benchmark trial, where the load parameter satisfies $x > 0$. Workload 1 performs $W_1(x) = (2x + 3)(3x + 1)$ million operations, Workload 2 performs $W_2(x) = (x + 4)(x + 2)$ million operations, and a background routine performs $W_3(x) = 8x + 1$ million operations. Which expression represents the total number of operations performed by the cluster, $W(x) = W_1(x) + W_2(x) + W_3(x)$, in millions, in completely factored form?

   • A) $(7x + 4)(x + 3)$
   • B) $(7x + 3)(x + 4)$
   • C) $(7x + 12)(x + 1)$
   • D) $(7x + 6)(x + 2)$
4. 4.

   A civil engineering firm designs an outdoor civic esplanade consisting of an adjoining trapezoidal pavilion and a rectangular promenade. The trapezoidal pavilion has parallel bases of lengths $3x - 1$ meters and $x + 5$ meters, with a perpendicular height of $2x + 4$ meters, where $x > 1$. Directly connected to the pavilion is the rectangular promenade, which has a length of $3x + 1$ meters and a width of $x + 2$ meters. Which expression represents the total combined ground area of the esplanade, in square meters, in completely factored form?

   • A) $(7x + 5)(x + 2)$
   • B) $(11x + 9)(x + 2)$
   • C) $7(x + 1)(x + 2)$
   • D) $(x + 3)(x + 2)$
5. 5.

   An orbital communications satellite generates electrical power using two deployed solar panel wings, where the operational parameter satisfies $x > 1$. Wing 1 generates an electrical power output modeled by $P_1(x) = (2x + 3)(x + 1)$ watts, while Wing 2 generates an output modeled by $P_2(x) = (x + 2)(x + 4)$ watts. During standard operations, the satellite's onboard flight instruments consume a continuous baseline power of $L(x) = x + 3$ watts. Which expression represents the net surplus power delivered to the satellite's storage batteries, $P_{\text{net}}(x) = P_1(x) + P_2(x) - L(x)$, in watts, in completely factored form?

   • A) $(3x + 2)(x + 4)$
   • B) $(3x + 8)(x + 1)$
   • C) $(3x - 4)(x - 2)$
   • D) $(3x + 4)(x + 2)$
6. 6.

   A precision engineering workshop fabricates a custom steel mounting bracket consisting of a central square plate of side length $2x + 3$ centimeters and two identical rectangular side tabs, each of length $2x + 3$ centimeters and width $x - 1$ centimeters. Before finishing, a rectangular mounting aperture measuring $3x - 1$ centimeters by $x + 1$ centimeters is cut out from the bracket, where $x > 1$. Which expression represents the remaining surface area of the bracket, in square centimeters, in completely factored form?

   • A) $(5x + 2)(x + 2)$
   • B) $(5x + 4)(x + 1)$
   • C) $(5x + 1)(x + 4)$
   • D) $(5x + 1)(x + 2)$
7. 7.

   An environmental engineering facility operates a multi-stage bio-filtration system to extract contaminants from industrial wastewater. During the primary filtration stage, the mass of contaminants captured is modeled by $P(x) = x^3 - 4x$ grams, and during the secondary stage an additional $Q(x) = 5x^2 - 20$ grams are neutralized, where $x > 2$. During the final cleaning cycle, $R(x) = 2x^2 - 8$ grams of particulate matter are flushed out, so the net total mass of contaminants processed is $M(x) = P(x) + Q(x) - R(x)$. Which expression represents the net total mass of contaminants processed, in grams, in completely factored form?

   • A) $(x - 2)^2(x + 3)$
   • B) $(x - 3)(x - 2)(x + 2)$
   • C) $(x - 2)(x + 2)(x + 3)$
   • D) $(x - 2)(x + 2)(x + 7)$
8. 8.

   An autonomous electric test vehicle completes a two-phase trial on a straight test track. In Phase 1, the vehicle travels at a constant speed of $2x + 1$ meters per second for $x + 2$ seconds. In Phase 2, it travels at a constant speed of $4x + 4$ meters per second for $x + 1$ seconds, where $x > 0$. Which expression represents the total distance traveled by the vehicle during the two-phase trial, in meters, in completely factored form?

   • A) $(2x + 3)(3x + 2)$
   • B) $(2x + 3)(6x + 5)$
   • C) $(6x + 1)(x + 2)$
   • D) $(3x + 1)(2x + 6)$
9. 9.

   A clean-technology enterprise manufactures modular filtration units. In a fiscal quarter, the total revenue, in thousands of dollars, generated from selling $x$ batches of filtration units is modeled by $R(x) = (2x + 5)(3x^2 + 4x + 1)$, while the total manufacturing cost, in thousands of dollars, is modeled by $C(x) = (x + 1)(2x^2 + 7x + 5)$, where $x > 1$. The company's net profit is defined as the difference between its total revenue and total manufacturing cost, $P(x) = R(x) - C(x)$. Which expression represents the company's net quarterly profit, in thousands of dollars, in completely factored form?

   • A) $2(x + 1)^2(4x + 5)$
   • B) $2x(2x + 1)(x + 5)$
   • C) $2x(x + 1)(2x + 5)$
   • D) $2(x + 1)^2(2x + 5)$
10. 10.

    A civil engineering laboratory designs two prototype sedimentation tanks, Tank A and Tank B, each constructed with a uniform vertical depth of $x$ meters, where $x > 1$. Tank A features a horizontal square base with an edge length of $2x + 5$ meters, whereas Tank B features a horizontal square base with an edge length of $x - 1$ meters. Which expression represents the difference between the volume of Tank A and the volume of Tank B, in cubic meters, in completely factored form?

    • A) $x(x + 3)(3x + 8)$
    • B) $x(x + 4)(3x + 4)$
    • C) $x(x + 6)(3x + 4)$
    • D) $x(x + 12)(3x + 2)$
11. 11.

    A precision manufacturing workshop cuts out four identical square corners, each with a side length of $x + 1$ centimeters, from a large square sheet of aluminum with a side length of $3x + 4$ centimeters, where $x > 1$. The remaining cross-shaped sheet is prepared to be folded into an open storage tray. Which expression represents the area of one face of this remaining cross-shaped sheet, in square centimeters, in completely factored form?

    • A) $(5x + 6)(x + 2)$
    • B) $(5x + 2)(x + 6)$
    • C) $(2x + 3)(4x + 5)$
    • D) $(5x + 12)(x + 1)$
12. 12.

    A manufacturing company produces open-top metal storage bins for industrial recycling. Each bin is designed in the shape of a rectangular prism with an open top, featuring a square base of side length $3x - 2$ decimeters and four vertical rectangular sides each of height $x + 4$ decimeters, where $x > 2$. Which expression represents the total exterior surface area of the bin in square decimeters, in completely factored form?

    • A) $2(3x - 2)(2x + 1)$
    • B) $2(3x - 2)(5x + 6)$
    • C) $7(3x - 2)(x + 2)$
    • D) $7(3x + 2)(x - 2)$
13. 13.

    Two adjacent rectangular greenhouse sections share a boundary along a common width. Section 1 has an area modeled by $3x^2 + 11x + 6$ square meters, and Section 2 has an area modeled by $2x^2 + 7x + 3$ square meters, where $x > 1$. If the shared width is represented by a non-constant binomial with integer coefficients, which expression represents the total combined length of the two sections in meters?

    • A) $5x + 5$
    • B) $4x + 7$
    • C) $5x + 7$
    • D) $5x + 3$
14. 14.

    A logistics warehouse monitors the daily cargo throughput of two automated loading bays. Bay 1 operates for $x + 7$ hours at an average transfer rate of $x + 2$ tons per hour, while Bay 2 operates for $x + 1$ hours at an average transfer rate of $x + 1$ tons per hour, where $x > 0$. The total daily cargo throughput is the sum of the amounts transferred by both bays. Which expression represents the total daily cargo throughput, in tons, in completely factored form?

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

    A landscape architect designs a community park containing a central open lawn, an equipment shed, and a paved walkway. The entire park is rectangular with a length of $3x + 4$ meters and a width of $2x + 5$ meters, where $x > 0$. The shed occupies a rectangular footprint of length $x + 3$ meters and width $x + 2$ meters, while the walkway covers an area of $x^2 + 4x + 2$ square meters. Which expression represents the remaining open lawn area of the park, in square meters, in completely factored form?

    • A) $2(2x + 3)(x + 2)$
    • B) $4(x + 1)(x + 3)$
    • C) $4(x + 1)(x + 7)$
    • D) $2(2x + 1)(x + 6)$
16. 16.

    A workshop manufactures modular wooden display cases whose total storage volume in cubic decimeters is modeled by $V(x) = 2x^3 + 7x^2 + 2x - 3$, where $x > 1$. The case consists of a lower storage compartment with a volume of $V_1(x) = 2x^3 + 3x^2 - 2x$ cubic decimeters and an upper open shelf, both sharing a uniform depth of $(2x - 1)$ decimeters. Which expression represents the front-facing area of the upper shelf in square decimeters?

    • A) $2x - 1$
    • B) $2x - 3$
    • C) $2x + 3$
    • D) $4x + 3$
17. 17.

    An urban planner designs a civic plaza by combining two adjacent parcels of land whose areas in square meters are modeled by $P(x) = 2x^2 + 9x - 6$ and $Q(x) = x^2 - x - 10$, where $x > 2$. The total combined area is then reconfigured into a single rectangular space with a width of $(x + 4)$ meters. Which expression represents the length of this reconfigured plaza in meters?

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

Answers and step-by-step solutions appear after you finish the test.