Simplifying and factoring linear expressions Transitional Reasoning Download PDF

Questions

1. 1.

   An agricultural processing plant stores liquid whey in two separate holding vats, Vat 1 and Vat 2. Vat 1 currently holds $7(6x + 5)$ liters of whey, and Vat 2 holds $3(6x + 1)$ liters of whey. A plant technician needs to equalize the volume of whey between the two vats by pumping a portion from Vat 1 into Vat 2. Which expression represents the volume of whey, in liters, that must be transferred from Vat 1 to Vat 2 in fully factored form?

   • A) $8(3x + 4)$
   • B) $6(2x + 3)$
   • C) $2(6x + 7)$
   • D) $4(3x + 4)$
2. 2.

   An exhibition coordinator organizes a rectangular display gallery that has a uniform width of 4 meters. The gallery is divided lengthwise into three themed pavilions: a robotics pavilion with an area of $6(5x + 2)$ square meters, an aerospace pavilion with an area of $4(3x + 1)$ square meters, and an energy pavilion with an area of $2(3x + 4)$ square meters. Which expression represents the total length of the gallery, in meters, in fully factored form?

   • A) $24(2x + 1)$
   • B) $6(2x + 1)$
   • C) $4(3x + 2)$
   • D) $3(4x + 2)$
3. 3.

   An architectural fabricator designs a decorative entryway sign in the shape of a symmetrical pentagon, formed by mounting an isosceles triangular crest directly on top of a rectangular base plate. The bottom horizontal edge of the rectangular base measures $6(2x + 4)$ centimeters, and each of its two vertical sides measures $3(3x + 2)$ centimeters. Each of the two slanted upper edges of the triangular crest measures $2(3x + 5)$ centimeters. Which expression represents the total outer perimeter of the sign, in centimeters, in fully factored form?

   • A) $2(27x + 40)$
   • B) $14(3x + 4)$
   • C) $2(18x + 23)$
   • D) $27x + 40$
4. 4.

   A conservation team plants two designated plots, Plot Alpha and Plot Beta, with native flora. Plot Alpha receives 3 rows of evergreen shrubs and 2 rows of flowering perennials, where each row of shrubs contains $4x + 7$ seedlings and each row of perennials contains $4x + 3$ seedlings. Plot Beta receives 2 rows of the same shrubs and 5 rows of the same perennials. Which expression represents the total number of seedlings planted across both plots in fully factored form?

   • A) $4(12x + 14)$
   • B) $8(6x + 7)$
   • C) $6(8x + 9)$
   • D) $16(3x + 4)$
5. 5.

   A science student uses a two-pan balance scale to find the mass of an unknown mineral specimen. On the left pan, the student places 5 identical cylindrical weights, each with a mass of $7x + 9$ grams, and 4 identical spherical weights, each with a mass of $3x + 4$ grams. On the right pan, the student places 2 of the same cylindrical weights, a calibration block with a mass of $9x + 3$ grams, and the mineral specimen. If the scale is in perfect equilibrium, which expression represents the mass, in grams, of the mineral specimen in fully factored form?

   • A) $4(6x + 10)$
   • B) $2(12x + 41)$
   • C) $8(3x + 5)$
   • D) $12(x + 2)$
6. 6.

   A cloud computing facility manages data backups across two server banks before transferring files to an archive. Server Bank 1 operates 3 storage units that each process $4(5x + 2)$ gigabytes of data, and Server Bank 2 operates 2 storage units that each process $3(6x + 5)$ gigabytes of data. Before transmission, an automated deduplication protocol eliminates $6(4x - 1)$ gigabytes of redundant data from the combined total. Which expression represents the final amount of data, in gigabytes, transmitted to the archive in fully factored form?

   • A) $12(6x + 5)$
   • B) $6(12x + 10)$
   • C) $24(3x + 2)$
   • D) $12(8x + 7)$
7. 7.

   An architect designs an outdoor amphitheater stage in the shape of an isosceles trapezoid. The shorter front boundary has a length of $3(2x + 4)$ meters, and the longer rear boundary has a length of $2(5x + 7)$ meters. Each of the two congruent slanted side boundaries has a length of $2(5x + 1)$ meters. Which expression represents the perimeter of the stage, in meters, in fully factored form?

   • A) $6(6x + 5)$
   • B) $2(13x + 14)$
   • C) $2(18x + 15)$
   • D) $4(9x + 7)$
8. 8.

   During a high school robotics challenge, a team scores points across three tasks by collecting 5 power cells worth $8x + 7$ points each, deploying 4 beacon links worth $6x + 5$ points each, and retrieving 3 telemetry probes worth $4x - 1$ points each. At the end of the round, the team receives a penalty deduction of $4(3x - 1)$ points for arena boundary violations. Which expression represents the team's final score in fully factored form?

   • A) $16(4x + 3)$
   • B) $8(11x + 6)$
   • C) $8(8x + 7)$
   • D) $4(19x + 13)$
9. 9.

   An alpine research team monitors the elevation of a field crew traversing a mountain pass. The crew departs from a base camp at an elevation of $4(5x + 6)$ meters, climbs $3(6x + 5)$ meters to a ridge, descends $2(4x - 3)$ meters into a saddle, and makes a final climb of $5(2x + 1)$ meters to reach an observation summit. Which expression represents the elevation of the observation summit, in meters, in fully factored form?

   • A) $10(4x + 5)$
   • B) $2(20x + 19)$
   • C) $2(28x + 19)$
   • D) $5(8x + 10)$
10. 10.

    A community arts center offers a seasonal membership pass that includes admission to 4 theater performances valued at $6x + 5$ dollars each, 3 gallery workshops valued at $4x - 1$ dollars each, and a voucher worth $2x + 7$ dollars. During an early-bird promotion, a discount of $8x + 4$ dollars is subtracted from the total value of each pass. A local company purchases 4 of these discounted passes for its employees. Which expression represents the total amount paid by the company, in dollars, in fully factored form?

    • A) $10(3x + 2)$
    • B) $8(15x + 14)$
    • C) $8(23x + 14)$
    • D) $40(3x + 2)$
11. 11.

    A framer constructs a rectangular wooden picture frame. The outer boundary of the frame has a length of $3(3x + 4)$ centimeters and a width of $2(x + 2)$ centimeters. The inner rectangular opening has a length of $4(x + 1)$ centimeters and a width of $x + 3$ centimeters. Which expression represents the difference between the outer perimeter and the inner perimeter of the frame, in centimeters, in fully factored form?

    • A) $3(2x + 3)$
    • B) $2(6x + 23)$
    • C) $2(16x + 23)$
    • D) $6(2x + 3)$
12. 12.

    A municipal botanical conservatory installs 4 identical climate-controlled terrarium zones. The conservatory receives a bulk shipment containing $7(12x + 15)$ kilograms of enriched substrate. Before filling the zones, the maintenance crew sets aside $3(4x + 3)$ kilograms of substrate for laboratory quality tests, and divides the remainder equally among the 4 zones. Which expression represents the mass of substrate, in kilograms, allocated to each terrarium zone in fully factored form?

    • A) $6(3x + 4)$
    • B) $24(3x + 4)$
    • C) $3(6x + 8)$
    • D) $6(3x + 16)$
13. 13.

    An urban planner designs a pedestrian plaza in the shape of an equilateral triangle with an initial perimeter of $15x + 36$ meters. To accommodate outdoor seating, the plan extends each of the three sides by $2x - 1$ meters. Which expression represents the new perimeter of the expanded plaza, in meters, in fully factored form?

    • A) $3(7x + 13)$
    • B) $3(7x + 9)$
    • C) $3(7x + 11)$
    • D) $3(3x + 13)$
14. 14.

    An automated guided vehicle (AGV) in a fulfillment center travels along a straight test track under the action of several horizontal forces. Two electric drive motors propel the vehicle forward with forces of $5(6x + 7)$ newtons and $3(4x + 5)$ newtons. At the same time, mechanical friction and incline drag oppose the forward motion with forces of $2(5x - 1)$ newtons and $4(2x + 4)$ newtons, respectively. Which expression represents the net forward force, in newtons, acting on the vehicle in fully factored form?

    • A) $4(15x + 16)$
    • B) $12(2x + 3)$
    • C) $6(4x + 6)$
    • D) $8(3x + 8)$
15. 15.

    A student council organizes a charity film screening to raise money for a community garden. The council collects $8(5x + 7)$ dollars in ticket sales and incurs expenses of $6(2x - 1)$ dollars for promotional posters and $2(4x + 5)$ dollars for refreshments. Which expression represents the council's net profit, in dollars, in fully factored form?

    • A) $4(5x + 13)$
    • B) $4(5x + 15)$
    • C) $2(10x + 26)$
    • D) $4(7x + 11)$
16. 16.

    An environmental monitoring station tracks the volume of water in a municipal conservation basin. Two automated intake channels, Channel A and Channel B, feed water into the basin at rates of $6(5x + 3)$ liters per minute and $2(4x + 9)$ liters per minute, respectively. At the same time, an overflow spillway releases water from the basin at a rate of $2(3x + 6)$ liters per minute. Which expression represents the net increase in the basin's water volume, in liters per minute, in fully factored form?

    • A) $4(8x + 6)$
    • B) $16(2x + 3)$
    • C) $4(11x + 12)$
    • D) $8(4x + 3)$
17. 17.

    Two community solar installations, Array Alpha and Array Beta, supply renewable electricity to a municipal microgrid. On a sunny day, Array Alpha operates $5$ identical solar panel banks that each generate $(6x + 8)$ kilowatt-hours, while Array Beta operates $2$ solar panel banks that each generate $(3x + 2)$ kilowatt-hours. Which expression represents how many more kilowatt-hours Array Alpha generates than Array Beta, in fully factored form?

    • A) $4(9x + 11)$
    • B) $4(6x + 11)$
    • C) $6(4x + 6)$
    • D) $12(2x + 3)$
18. 18.

    A graphic designer creates a custom rectangular banner with a perimeter of $22x + 38$ centimeters. The width of the banner is given by the expression $2(3x + 4) + (x - 1)$ centimeters. Which expression represents the length of the adjacent side, in centimeters, in fully factored form?

    • A) $15(x + 3)$
    • B) $2(2x + 13)$
    • C) $2(2x + 6)$
    • D) $4(x + 3)$
19. 19.

    A school robotics team designs an obstacle course using modular track sections. The initial layout has a total length of $8(3x + 5)$ centimeters. During testing, the team removes a curved loop measuring $3(2x + 4)$ centimeters and an incline section measuring $2(3x + 2)$ centimeters. Which expression represents the remaining length, in centimeters, of the track layout in fully factored form?

    • A) $4(9x + 14)$
    • B) $4(3x + 14)$
    • C) $6(2x + 4)$
    • D) $12(x + 2)$
20. 20.

    A landscape designer is installing border edging around three distinct sections of a botanical garden. The first section requires $4(2x + 3)$ meters of edging, the second section requires $2(5x - 1)$ meters, and the third section requires $3(2x - 2)$ meters. The designer needs to express the total length of edging in simplified and fully factored form. Which expression represents the total length, in meters, of border edging required?

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