Which expression represents "the square of a number n, decreased by 7"?
Hint
Find "the square of a number" first. Then decide what happens to that result.
Step-by-step solution
"The square of a number n" is n^2.
"Decreased by 7" means subtract 7 from that result.
Start with n^2 and take away 7: n^2 - 7.
Answer: n^2 - 7
Check: Let n = 5. The square of 5 is 25, and 25 decreased by 7 is 18. The expression gives 5^2 - 7 = 18 ✓. By contrast (5 - 7)^2 = 4, which does not match the words.
Tip: Read where the word "square" points. Here it points to the number alone, so only n is squared. If the words were "the square of 7 less than n," the whole group (n - 7) would be squared.
Question 2 of 36
No calculator
The length of a rectangular rug is 4 feet more than its width, w. Which expression represents the area of the rug, in square feet?
Hint
Write the length in terms of w. Area = length × width.
Step-by-step solution
Width = w.
"4 feet more than its width" means length = w + 4.
Area = length × width = w(w + 4).
Distribute: w \cdot w + w \cdot 4 = w^2 + 4w.
Answer: w^2 + 4w
Check: If w = 3, the rug is 3 ft by 7 ft, so its area is 21 ft². The expression gives 3^2 + 4(3) = 9 + 12 = 21 ✓.
Tip: When you distribute, the outside factor multiplies every term inside the parentheses. Area of a shape with two variable sides will have a squared term.
Question 3 of 36
No calculatorFill in
A square tile has sides of length (x + 5) centimeters. Its area can be written as x^2 + bx + 25 square centimeters.
What is the value of b?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The area of a square is (side)². Write (x + 5)^2 as (x + 5)(x + 5) and multiply all four pairs.
Step-by-step solution
Area of a square = side \times side = (x + 5)(x + 5).
FOIL: First x \cdot x = x^2; Outer x \cdot 5 = 5x; Inner 5 \cdot x = 5x; Last 5 \cdot 5 = 25.
Combine the middle terms: 5x + 5x = 10x.
Area = x^2 + 10x + 25, so b = 10.
Answer: b = 10
Check: Let x = 1. The side is 6 cm and the area is 36 cm². The polynomial gives 1 + 10 + 25 = 36 ✓. With b = 5 you would get 31, which is wrong.
Tip: The pattern (a + b)^2 = a^2 + 2ab + b^2 means the middle coefficient is twice the product, so the middle term is 2 \cdot 5 \cdot x = 10x.
Question 4 of 36
No calculator
Which expression represents "three times the cube of a number n, increased by twice the number"?
Hint
Break the phrase into pieces. "The cube of n" comes first; then "three times" that. "Twice the number" means 2 times n.
Step-by-step solution
"The cube of a number n" is n^3.
"Three times the cube" multiplies that by 3: 3n^3. (Only the cube is tripled, not n before cubing.)
"Twice the number" is 2 \cdot n = 2n.
"Increased by" means add: 3n^3 + 2n.
Answer: 3n^3 + 2n
Check: Try n = 2: the cube is 8, three times that is 24, twice the number is 4, and 24 + 4 = 28. The expression gives 3(8) + 2(2) = 28 ✓.
Tip: In word phrases, "three times the cube of n" is 3n^3, while "the cube of three times n" is (3n)^3 = 27n^3. Watch which operation is applied first.
Question 5 of 36
No calculator
A rectangular dog run is x feet wide. Its length is 6 feet more than 3 times its width.
Which expression represents the area of the dog run, in square feet?
Hint
Write the length in terms of x first. Then use area = length × width and distribute.
Step-by-step solution
Width = x.
"6 more than 3 times the width": length = 3x + 6.
Area = \text{length} \times \text{width} = x(3x + 6).
Distribute x to both terms: x \cdot 3x = 3x^2 and x \cdot 6 = 6x.
Area = 3x^2 + 6x.
Answer: 3x^2 + 6x square feet
Check: If x = 4, the run is 4 ft by 3(4) + 6 = 18 ft, so area = 72 ft². Expression: 3(16) + 6(4) = 48 + 24 = 72 ✓.
Tip: Area uses multiplication, so expect an x^2 term when both sides contain x. An answer with only x terms is probably a perimeter.
Question 6 of 36
No calculator
Three integers come one right after another (for example 7, 8, 9). The least of them is called n. Which expression gives their total when the three are added?
Hint
Write the three integers in terms of n first. Consecutive integers go up by 1.
Step-by-step solution
Consecutive integers differ by 1, so the three integers are n, n + 1, and n + 2.
Sum: n + (n + 1) + (n + 2).
Combine like terms: n + n + n = 3n and 1 + 2 = 3.
Sum = 3n + 3.
Answer: 3n + 3
Check: With n = 7: 7 + 8 + 9 = 24, and 3(7) + 3 = 24 ✓.
Tip: Use +1 steps for consecutive integers, and +2 steps for consecutive even or odd integers. "Sum" means add; "product" means multiply.
Question 7 of 36
Drop-down
A theater sells 400 tickets per show at $10 each. For each $1 increase in ticket price, it expects to sell 15 fewer tickets. Let x be the number of $1 increases.
The theater's revenue per show, in dollars, is represented by .
Choose an answer from each menu.
Hint
Revenue = (price per ticket) × (number of tickets). Write each one in terms of x first.
Step-by-step solution
Each $1 increase raises the price by 1, so after x increases the price is 10 + x dollars.
Each increase loses 15 tickets, so after x increases the number sold is 400 - 15x.
Revenue = price × number sold = (10 + x)(400 - 15x).
Answer: (10 + x)(400 - 15x)
Check: Let x = 2. Price $12, tickets 400 - 30 = 370, revenue 12 \cdot 370 = 4440. The expression gives (12)(370) = 4440 ✓.
Tip: "For each" means the change is multiplied by the variable. Then revenue always multiplies the two pieces; never add them.
Question 8 of 36
A rectangular pool measures 30 feet by 20 feet. A walkway of uniform width x feet surrounds it, as shown.
Which expression represents the total area of the pool and walkway together, in square feet?
Hint
The walkway is on both ends of each side. What are the outer length and width?
Step-by-step solution
The walkway adds x on the left andx on the right, so the outer length is 30 + 2x.
It also adds x on the top and bottom, so the outer width is 20 + 2x.
Check: If x = 5, the outer rectangle is 40 \times 30 = 1200 ft². The expression gives 4(25) + 100(5) + 600 = 100 + 500 + 600 = 1200 ✓.
Tip: A uniform border adds twice its width to each dimension. Sketch the figure and label each outer side before you multiply.
Question 9 of 36
Fill in
A box maker starts with a 20-inch by 12-inch sheet of cardboard. An x-inch square is cut from each corner and the sides are folded up to make an open box.
Write an expression for the volume of the box. Then find the volume, in cubic inches, when x = 3.
in³
Type your answer in the box. Fractions like 3/4 are fine.
Hint
After folding, what are the length, width and height of the box? Each side of the base loses a square at both ends.
Step-by-step solution
Height: the folded-up flaps are x inches tall, so h = x.
Length of the base: 20 - 2x (an x is removed at each end).
Width of the base: 12 - 2x.
Volume V = x(20 - 2x)(12 - 2x), which simplifies to 4x^3 - 64x^2 + 240x.
Check: Let x = 2. The side is 5 m and the area is 25 m². (5)^2 = 25 ✓, 5 \cdot 5 = 25 ✓, 4 + 12 + 9 = 25 ✓. But x^2 + 9 = 13, 2x + 6 = 10, x^2 + 3x + 9 = 19 — all wrong.
Tip: When you square a sum, the middle term is twice the product of the two parts (2 \cdot 3 \cdot x = 6x).
Question 11 of 36
No calculator
Let n be the smaller of two consecutive odd integers. Which expression represents the product of the two integers?
Hint
Odd numbers such as 7, 9, 11 are how far apart?
Step-by-step solution
Consecutive odd integers are 2 apart (for example 7 and 9), so if the smaller one is n, the larger is n + 2.
"Product" means multiply: n(n + 2).
Distribute: n \cdot n + n \cdot 2 = n^2 + 2n.
Answer: n^2 + 2n
Check: Let n = 7. The integers are 7 and 9, and 7 \cdot 9 = 63. The expression gives 49 + 14 = 63 ✓.
Tip: Consecutive integers go up by 1; consecutive odd or even integers go up by 2. Test your expression with a real pair of numbers.
Question 12 of 36
Drag and drop
Drag each expression into the box for the phrase it represents. One expression is not used.
the square of the sum of x and 4
the sum of the square of x and 4
4 times the square of x
Tap a tile, then tap a box to place it.
Hint
Find the words that come right after "the square of." That is the part that gets squared.
Step-by-step solution
"The square of the sum of x and 4": the sum is x + 4, and the whole sum is squared: (x + 4)^2.
"The sum of the square of x and 4": square x first, then add 4: x^2 + 4.
"4 times the square of x": square x, then multiply by 4: 4x^2.
The leftover tile, (4x)^2 = 16x^2, would be "the square of 4 times x."
Answer: (x + 4)^2; x^2 + 4; 4x^2
Check: Let x = 2. "Square of the sum of 2 and 4" = 6^2 = 36, and (2 + 4)^2 = 36 ✓. "Sum of the square of 2 and 4" = 4 + 4 = 8 ✓. "4 times the square of 2" = 4 \cdot 4 = 16 ✓, while (4 \cdot 2)^2 = 64 ✗.
Tip: Parentheses in algebra play the role of "the sum of" or "the product of" in words. Translate the grouping first.
Question 13 of 36
A triangular flower bed has sides of length x, x^2 - 1, and 2x + 5 feet.
Which expression represents the perimeter of the flower bed, in feet?
Hint
Perimeter is the sum of all sides. Combine only like terms: x^2 with x^2, x with x, numbers with numbers.
Step-by-step solution
Perimeter = sum of the three sides: x + (x^2 - 1) + (2x + 5).
Group like terms:
x^2 terms: x^2
x terms: x + 2x = 3x
constants: -1 + 5 = 4
Perimeter = x^2 + 3x + 4.
Answer: x^2 + 3x + 4 feet
Check: With x = 3: sides are 3, 8, and 11 ft, total 22 ft. Expression: 9 + 9 + 4 = 22 ✓.
Tip:x^2 and x are not like terms, so they never combine. Keep track of the sign on each constant (here -1).
Question 14 of 36
A bakery models its weekly revenue from x batches of bread as R = 4x^2 + 30x dollars. Its weekly cost is C = x^2 + 6x + 200 dollars.
Which expression represents the weekly profit, R - C, in dollars?
Hint
Put the cost in parentheses before subtracting. The minus sign applies to every term of the cost.
Distribute the minus sign: 4x^2 + 30x - x^2 - 6x - 200.
Combine like terms:
4x^2 - x^2 = 3x^2
30x - 6x = 24x
constant: -200
Profit = 3x^2 + 24x - 200.
Answer: 3x^2 + 24x - 200 dollars
Check: With x = 10: R = 400 + 300 = 700, C = 100 + 60 + 200 = 360, profit = 340. Expression: 300 + 240 - 200 = 340 ✓.
Tip: Subtracting a polynomial means adding the opposite of each term. A cost's fixed part (here $200) always ends up negative in the profit.
Question 15 of 36
A community theater models its revenue per show as
R = (12 + x)(250 - 10x)
where x is the number of $1 increases in the ticket price.
What does the factor 250 - 10x represent?
Hint
Revenue = (price per ticket) × (number of tickets). Which factor starts at the original price?
Step-by-step solution
Revenue is always (price per ticket) × (number of tickets sold).
The factor 12 + x starts at 12 and goes up $1 for each increase, so it is the ticket price in dollars.
The factor 250 - 10x starts at 250 and goes down by 10 for each $1 increase, so it is the number of tickets sold.
So the theater sells 250 tickets at $12, and each $1 increase costs it 10 ticket sales.
Answer: the number of tickets sold per show after x price increases
Check: With x = 0: price $12, tickets 250, revenue $3,000. With x = 2: price $14, tickets 230, revenue 14 \cdot 230 = \$3{,}220. The second factor dropped by 20 = 2 × 10 ✓.
Tip: In revenue models, the factor containing the dollar amount is the price; the factor with the large count is the quantity. The coefficient of x in each factor is the change per step.
Question 16 of 36
A rectangular piece of plywood is (2x + 6) inches long and (x + 4) inches wide. A square with sides of x inches is cut out of one corner, as shown.
Which expression represents the area of the remaining plywood, in square inches?
Hint
Remaining area = area of the whole rectangle − area of the square that was removed.
Check: With x = 2: the board is 10 \times 6 = 60 in², the cut-out is 4 in², leaving 56 in². Expression: 4 + 28 + 24 = 56 ✓.
Tip: For any "shape with a piece removed," write Big − Small first, then simplify. Multiplying two binomials needs all four FOIL products.
Question 17 of 36
Fill in
Each side of a square rug measures (2x - 3) feet. When the rug's area is multiplied out into the form ax^2 + bx + c square feet, what number is b?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Area of a square = side × side. Write (2x - 3)(2x - 3) and use all four FOIL products.
Step-by-step solution
Area = (2x - 3)^2 = (2x - 3)(2x - 3).
FOIL:
First: 2x \cdot 2x = 4x^2
Outer: 2x \cdot (-3) = -6x
Inner: (-3) \cdot 2x = -6x
Last: (-3)(-3) = 9
Combine: 4x^2 - 12x + 9.
So a = 4, b = -12, c = 9.
Answer: b = -12
Check: With x = 5: side = 7 ft, area = 49 ft². Expression: 4(25) - 12(5) + 9 = 100 - 60 + 9 = 49 ✓.
Tip: Shortcut for a squared binomial: (p - q)^2 = p^2 - 2pq + q^2. The middle coefficient is -2 \cdot 2 \cdot 3 = -12. Do not forget its negative sign.
Question 18 of 36
Fill in
A rectangular solar panel is w feet wide. Its length is 2 feet more than 3 times its width.
Write a polynomial for the area of the panel. Then find the area, in square feet, when w = 4.
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Length = 3w + 2. Multiply by the width, then substitute w = 4.
Tip: Square the variable before multiplying by its coefficient: 3w^2 at w = 4 is 3 \cdot 16 = 48, not 12^2 = 144. Checking with the actual length and width catches this error.
Question 19 of 36
A bike rental shop rents 60 bikes a day at $25 each. For each $5 increase in the daily price, it expects to rent 4 fewer bikes. Let x be the number of $5 increases.
Which polynomial represents the shop's daily rental revenue, in dollars?
Hint
Price = 25 + 5x (each step is $5, not $1). Bikes rented = 60 - 4x. Multiply and combine like terms.
Tip: The size of each price step goes into the price factor: $5 steps give 25 + 5x. Using 25 + x would mean $1 steps.
Question 20 of 36
Drag and drop
Let n be an even integer. Drag each expression into the box for the phrase it represents. One expression is not used.
the next even integer after n
the product of n and the next integer
the sum of n and the next even integer
Tap a tile, then tap a box to place it.
Hint
Even integers are 2 apart (4, 6, 8). The very next integer is only 1 more.
Step-by-step solution
Next even integer after n: even numbers are 2 apart, so it is n + 2.
Product of n and the next integer: the next integer (any kind) is n + 1. Product: n(n + 1) = n^2 + n.
Sum of n and the next even integer:n + (n + 2) = 2n + 2.
The leftover tile n + 1 is the next integer, but no box asks for that by itself.
Answer: n + 2; n^2 + n; 2n + 2
Check: Let n = 6. Next even: 8, and 6 + 2 = 8 ✓. Product with next integer: 6 \cdot 7 = 42, and 36 + 6 = 42 ✓. Sum with next even: 6 + 8 = 14, and 2(6) + 2 = 14 ✓.
Tip: Pick a real number for n and test each tile. It is the fastest way to sort expressions.
Question 21 of 36
Fill in
A farmer is building a rectangular pen against the side of a barn, as shown. The barn forms one side, so no fence is needed there. Each side touching the barn is x feet, and the side opposite the barn is (3x - 4) feet.
Write an expression for the total length of fence. How many feet of fence are needed when x = 12?
ft
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Only three sides need fence. Add their lengths, combine like terms, and then substitute.
Step-by-step solution
The fenced sides are: two sides of x feet and one side of (3x - 4) feet. The barn side gets no fence.
Total fence: x + x + (3x - 4) = 5x - 4.
Substitute x = 12: 5(12) - 4 = 60 - 4 = 56.
Answer: 56 feet
Check: At x = 12 the sides are 12 ft, 12 ft and 3(12) - 4 = 32 ft. 12 + 12 + 32 = 56 ft ✓.
Tip: Read carefully which sides need material. A full perimeter would be 2x + 2(3x - 4) = 8x - 8, which would give 88 ft, too much fence.
Question 22 of 36
The cost, in dollars, of printing x custom T-shirts is modeled by
C = 0.02x^2 + 4x + 300
What does the 300 represent?
Hint
What is the cost when x = 0?
Step-by-step solution
Substitute x = 0: C = 0.02(0)^2 + 4(0) + 300 = 300.
So $300 is the cost even when no shirts are printed.
In business that is called a fixed cost (setup of screens, equipment, design fee).
The terms with x (0.02x^2 + 4x) are the costs that grow as more shirts are printed.
Answer: a fixed setup cost of $300 that is paid no matter how many shirts are printed
Check: Printing 1 shirt costs 0.02 + 4 + 300 = \$304.02; printing 10 costs 2 + 40 + 300 = \$342. The $300 is in every total, while the rest changes with x ✓.
Tip: In a polynomial model, the constant term is the value when the input is 0. Coefficients of x describe how the output changes.
Question 23 of 36
A farm stand sells 120 baskets of strawberries a week at $8 per basket. The owner estimates that for each $1 increase in price, she will sell 6 fewer baskets per week. Let x be the number of $1 increases.
Which polynomial, in simplified form, represents the weekly revenue in dollars?
Hint
Write the price and the number of baskets in terms of x, multiply them, and combine like terms carefully.
Step-by-step solution
Price per basket: 8 + x dollars.
Baskets sold: 120 - 6x.
Revenue = (8 + x)(120 - 6x).
FOIL: First 8 \cdot 120 = 960; Outer 8 \cdot (-6x) = -48x; Inner x \cdot 120 = 120x; Last x \cdot (-6x) = -6x^2.
Check: At x = 20: revenue = -800 + 1600 = \$800, cost = 200 + 150 = \$350, and 800 - 350 = \$450 ✓.
Tip: Revenue first, cost second. Reversing the order gives the negative of the profit.
Question 26 of 36
A wall is (x + 8) feet long and (x + 5) feet tall. It has a rectangular window that is x feet wide and 3 feet tall. Only the wall around the window will be painted.
Which expression represents the area to be painted, in square feet?
Hint
Painted area = wall area − window area. The window's area is x \cdot 3.
Check: With x = 3: the base is 14 \times 8 and the height is 3, so V = 336 in³. Formula: 3(14)(8) = 336 ✓. The factor 20 - 2(3) = 14 is indeed the length.
Tip: In a factored model, each factor usually stands for one measurable quantity. Match the starting number (20) with the part of the story it came from.
Question 31 of 36
A sheet of metal measures 24 inches by 18 inches. A square with side x inches is cut from each corner, and the sides are folded up to form an open-top tray.
Which polynomial represents the volume of the tray, in cubic inches?
Hint
Find the length, width and height after folding. Multiply the two binomials first, then multiply by the height.
Step-by-step solution
Length = 24 - 2x, width = 18 - 2x (a square is removed at both ends of each side).
Check: At x = 3 the plate is 11 \times 11 = 121 cm², the two holes are 2 \times 9 = 18 cm², and 121 - 18 = 103 cm² ✓.
Tip: Count the cut-outs. "Two holes" means subtract the hole area twice. Checking with real numbers is a quick way to confirm your polynomial.
Question 34 of 36
A rectangular garden is w feet wide, and its length is 5 feet more than twice its width. The owner plans to make the garden 3 feet longer and 3 feet wider.
Which expression represents the increase in the garden's area, in square feet?
Hint
Write the old area and the new area as polynomials. The increase is new − old.
Step-by-step solution
Old dimensions: width w, length 2w + 5. Old area = w(2w + 5) = 2w^2 + 5w.
New dimensions: width w + 3, length 2w + 5 + 3 = 2w + 8.
New area = (w + 3)(2w + 8) = 2w^2 + 8w + 6w + 24 = 2w^2 + 14w + 24.
Increase = new - old = (2w^2 + 14w + 24) - (2w^2 + 5w) = 9w + 24.
Answer: 9w + 24 square feet
Check: With w = 4: old garden 4 \times 13 = 52 ft²; new garden 7 \times 16 = 112 ft²; increase 60 ft². Expression: 9(4) + 24 = 60 ✓.
Tip: An "increase" or "change" is always a subtraction: new minus old. The w^2 terms cancel here, a good sign your setup is right.
Question 35 of 36
Select all
A concert hall sells 500 tickets at $30 each. For each $2 decrease in price, it expects to sell 40 more tickets. Let x be the number of $2 decreases.
Select all expressions that represent the revenue, in dollars.
Select all that apply.
Hint
Price = 30 - 2x, tickets = 500 + 40x. Revenue is their product, in any equivalent form.
Tip: A price decrease means - in the price factor and + in the quantity factor. Equivalent forms all give the same value for any x: test one.
Question 36 of 36
A rectangular pool is 25 feet long and 10 feet wide. A deck of uniform width x feet will be built along both 25-foot sides and one 10-foot side of the pool (not the other 10-foot side).
Which expression represents the area of the deck alone, in square feet?
Hint
Sketch it. The deck adds 2x to the width (both long sides) but only x to the length (one end).
Step-by-step solution
The deck runs along both 25-ft sides, so the total width is 10 + 2x.
It runs along only one 10-ft end, so the total length is 25 + x.
