Square 3 first (3 \cdot 3), then multiply 3 \cdot 3 for the middle term, then add and subtract.
Step-by-step solution
Substitute in parentheses: (3)^2 + 3(3) - 4.
Power: (3)^2 = 3 \cdot 3 = 9.
Multiply: 3(3) = 9.
Combine: 9 + 9 - 4 = 14.
The value is 14.
Check: on the calculator: 3x²+3×3−4enter → 14 ✓
Tip:x^2 is x times itself, not x times 2. At x = 3 these give 9 and 6 — very different results. And 3x means 3 times x; never write the digits side by side as "33".
Question 2 of 36
No calculatorFill in
Evaluate 2x^2 - 5 when x = -3.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Square -3 first. A negative number times itself is positive. Then multiply by 2.
Step-by-step solution
Substitute in parentheses: 2(-3)^2 - 5.
Power first: (-3)^2 = (-3)(-3) = 9.
Multiply: 2 \cdot 9 = 18.
Subtract: 18 - 5 = 13.
The value is 13.
Check: on the calculator: 2×((−)3)x²−5enter → 13 ✓
Tip: two common wrong answers are -23 (from (-3)^2 = -9) and 31 (from (2 \cdot -3)^2 = 36, squaring the coefficient too). Square only the base, and a squared negative is positive.
Question 3 of 36
No calculator
What is the value of -x^2 + 4x when x = -2?
Hint
In -x^2, square x first, then take the opposite. Watch the sign of 4x too.
Step-by-step solution
Substitute in parentheses: -(-2)^2 + 4(-2).
Power: (-2)^2 = 4, so -(-2)^2 = -4.
Multiply: 4(-2) = -8.
Add: -4 + (-8) = -12.
The value is -12.
Check:-x^2 is never positive, and 4x is negative when x is negative, so the answer must be negative. Only -12 and -4 are possible, and -4 - 8 = -12 ✓
Tip: the minus sign in -x^2 is applied after squaring. Think of -x^2 as -1 \cdot x^2.
Question 4 of 36
No calculator
If x = 3, what number does x^4 - 3x^2 + 2x equal?
Hint
Find each power first: 3^4 and 3^2. Then multiply by the coefficients, then add and subtract from left to right.
Check: the polynomial factors as (2x - 1)(3x - 1). At x = \tfrac{1}{2} the first factor is 2 \cdot \tfrac{1}{2} - 1 = 0, so the product is 0 ✓
Tip: an answer of 0 is allowed. It means x = \tfrac{1}{2} is a zero (root) of the polynomial. Squaring a fraction squares the top and the bottom.
Question 6 of 36
No calculator
Substitute x = -2 into x^3 - 2x^2 + 5. What is the result?
Hint
(-2)^3 has three negative factors, and (-2)^2 has two. Decide each sign before you combine.
Step-by-step solution
Substitute with parentheses: (-2)^3 - 2(-2)^2 + 5.
Cube: (-2)^3 = (-2)(-2)(-2) = -8 (odd number of negatives → negative).
Square: (-2)^2 = (-2)(-2) = 4 (even number of negatives → positive).
Multiply: -2 \cdot 4 = -8.
Combine: -8 - 8 + 5 = -11.
The value is -11.
Check: factor out x^2 from the first two terms: x^2(x - 2) + 5 = 4(-2 - 2) + 5 = -16 + 5 = -11 ✓
Tip: always put a negative input in parentheses when you substitute. An even power of a negative number is positive; an odd power is negative.
Question 7 of 36
No calculator
The table shows some values of p(x) = 2x^2 - x - 3.
x
-2
-1
0
1
2
p(x)
7
0
-3
?
3
What is the missing value p(1)?
Hint
Do not guess from the pattern. Substitute x = 1 into the formula.
Step-by-step solution
Substitute x = 1: p(1) = 2(1)^2 - (1) - 3.
Power: (1)^2 = 1, so 2(1)^2 = 2.
Combine: 2 - 1 - 3 = -2.
The missing value is p(1) = -2.
Check: confirm a known entry to make sure you are using the formula correctly: p(2) = 2(4) - 2 - 3 = 3 ✓ matches the table, and p(-1) = 2(1) + 1 - 3 = 0 ✓.
Tip: a polynomial table is not always symmetric around x = 0. Here p(-1) = 0 but p(1) = -2, because the -x term changes sign when x changes sign.
Question 8 of 36
A ball is thrown upward from a balcony. Its height, in feet, after t seconds is given by
h = -16t^2 + 64t + 5
What is the height of the ball after 1.5 seconds?
Hint
Square 1.5 first, then multiply by -16. Do not square the -16.
Tip: factoring out the variable, p(120 - 2p), is a handy way to double-check a polynomial value. Also remember: in -2p^2 only p is squared, and the result is then multiplied by -2.
Question 12 of 36
An open box is made from a 10-inch by 8-inch sheet of cardboard by cutting an x-inch square from each corner and folding up the sides.
The volume of the box, in cubic inches, is V = x(10 - 2x)(8 - 2x). What is the volume when x = 1.5?
Hint
Find each factor first: the height x, the length 10 - 2x and the width 8 - 2x. Then multiply all three.
Check: the formula V = lwh (on the formula sheet) gives 7 \times 5 \times 1.5 = 52.5 ✓. The length and width each lose 2x because a square is cut from both ends.
Tip: with a product of factors, evaluate each factor separately, then multiply. This is much safer than expanding the polynomial first.
Question 13 of 36
Select all
Select all polynomials that have a value of 0 when x = -1.
Select all that apply.
Hint
Use (-1)^2 = 1 and (-1)^3 = -1. Evaluate each choice carefully.
Tip: in -3x^2 the square is positive (16), but the coefficient -3 is still negative, so the term is negative. Square first, then apply the coefficient's sign.
Question 16 of 36
Fill in
Evaluate 3m^2n - mn^2 + 4 when m = 2 and n = -3.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write each term with the numbers in parentheses: 3(2)^2(-3), then (2)(-3)^2. Find the powers first.
Check: add the faces directly. Top and bottom: 2 \cdot 4.5^2 = 40.5. Four sides, each 4.5 \times 5 = 22.5: 4 \cdot 22.5 = 90. Total: 40.5 + 90 = 130.5 ✓
Tip: a formula given in the problem is all you need; you do not have to know where it comes from. Substitute carefully and let the calculator square the decimal.
Question 20 of 36
The population of a town, in thousands, t years after 2010 is modeled by
P = 0.2t^2 + 3t + 50
By how many thousand people does the model say the population increased from 2015 (t = 5) to 2020 (t = 10)?
Hint
Find P at t = 10 and at t = 5 separately, then subtract. Do not plug in 10 - 5.
Tip: "work backwards" problems become a simple linear equation once you substitute. Simplify the powers and signs first, then solve for the unknown coefficient.
Question 22 of 36
Drop-down
Complete the table of values for q(x) = x^3 - x^2 - 4.
x
-1
0
2
3
q(x)
-4
0
Choose an answer from each menu.
Hint
For x = -1: (-1)^3 = -1 and (-1)^2 = 1. For x = 3: 3^3 = 27 and 3^2 = 9.
(-1)^3 = -1 (odd power of a negative is negative).
(-1)^2 = 1, and the minus sign in front makes the term -1.
x = 3: q(3) = 3^3 - 3^2 - 4 = 27 - 9 - 4 = 14.
The missing entries are q(-1) = -6 and q(3) = 14.
Check: confirm a given entry with the same method: q(2) = 8 - 4 - 4 = 0 ✓ and q(0) = 0 - 0 - 4 = -4 ✓.
Tip: in -x^2, the square happens before the minus sign: -(-1)^2 = -(1) = -1. Write the input in parentheses every time.
Question 23 of 36
Select all
Which polynomials equal 10 at x = 2? Choose every polynomial that works.
Select all that apply.
Hint
Evaluate every choice at x = 2; more than one is correct. Use 2^2 = 4 and 2^3 = 8.
Step-by-step solution
Useful facts: 2^2 = 4 and 2^3 = 8.
x^3 + 2 = 8 + 2 = 10 ✓
x^3 + 4 = 8 + 4 = 12 ✗
x^2 - 3x + 12 = 4 - 6 + 12 = 10 ✓
3x^2 - 26 = 3(4) - 26 = 12 - 26 = -14 ✗
x^2 - 3x = 4 - 6 = -2 ✗
2x^2 + x = 2(4) + 2 = 8 + 2 = 10 ✓
The polynomials equal to 10 at x = 2 are x^3 + 2, x^2 - 3x + 12 and 2x^2 + x.
Check: each wrong choice gives 10 only through a typical error: x^3 + 4 with x^3 read as 3x; 3x^2 - 26 with (3x)^2 = 36; x^2 - 3x with the minus sign ignored (4 + 6). Recheck: 8 + 2, 4 - 6 + 12 and 8 + 2 are all 10 ✓
Tip: on a select-all question, test every choice. Do not stop after you find one that works.
Question 24 of 36
Fill in
A diver jumps from a platform. Her height above the water, in feet, t seconds after she jumps is
h = -16t^2 + 8t + 32
What is her height above the water after 1.25 seconds?
feet
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find (1.25)^2 first. Then multiply by -16, add 8 \times 1.25, and add 32.
Step-by-step solution
Substitute t = 1.25: h = -16(1.25)^2 + 8(1.25) + 32.
Tip: the constant (32) is the starting height at t = 0. A reasonable answer must be less than the highest point and not negative while she is still in the air.
Question 25 of 36
Fill in
For the polynomial p(x) = x^2 + kx - 6, the value p(3) = 12.
What is the value of k?
k =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
p(3) means substitute x = 3. Set the result equal to 12 and solve for k.
Check: the garden is x by (x + 4), so 5 \times 9 = 45 square feet ✓
Tip: when a question asks which input gives a certain output, evaluating the choices is often fastest. Start with a middle-sized choice so you know whether to go up or down.
Question 30 of 36
A company designs cylindrical cans whose height is always 3 inches more than the radius. The volume of a can with radius r inches is
V = \pi r^3 + 3\pi r^2
What is the volume, in cubic inches, of a can with a radius of 4 inches? Leave your answer in terms of \pi.
Hint
Find 4^3 and 4^2 first. Then add the two terms; both contain \pi, so you can add their coefficients.
Tip: choose the order that keeps numbers small: 9 \cdot \tfrac{4}{9} cancels to 4 right away. On the calculator, use the n/d key and put -\tfrac{2}{3} in parentheses before squaring.
Question 32 of 36
Drag and drop
Drag a value into each empty box to complete the table for f(x) = -x^2 + 3x + 1. Not every tile is used.
x
-2
0
2
4
f(x)
Tap a tile, then tap a box to place it.
Hint
-x^2 means "square x, then take the opposite". So -x^2 is negative (or zero) for every x.
Step-by-step solution
x = -2: -(-2)^2 + 3(-2) + 1 = -4 - 6 + 1 = -9.
x = 0: -0 + 0 + 1 = 1.
x = 2: -(2)^2 + 3(2) + 1 = -4 + 6 + 1 = 3.
x = 4: -(4)^2 + 3(4) + 1 = -16 + 12 + 1 = -3.
The table reads -9, 1, 3, -3.
Unused tiles: -1 comes from reading -(-2)^2 as +4; 29 comes from reading -(4)^2 as +16.
Check: the graph of f is a downward parabola with its top near x = 1.5, so values should fall off on both sides: 1 \to 3 near the top, then down to -3 and -9 ✓
Tip: the exponent acts before the negative sign: -x^2 = -(x^2), never (-x)^2.
Question 33 of 36
A company's weekly profit, in dollars, from making x chairs is
P = -0.5x^2 + 40x - 350
How much more profit does the company make in a week when it makes 40 chairs than when it makes 30 chairs?
Hint
Evaluate the profit at x = 40 and at x = 30, then subtract.
Check: on the calculator, (−).5×40x²+40×40−350enter → 450, and the same with 30 → 400 ✓
Tip: "how much more" questions need two evaluations and a subtraction. Choices like $450 (just one value) and $850 (the sum) are traps for stopping early.
Question 34 of 36
Two print shops price a custom order of x posters differently:
Shop A: C_A = 0.02x^2 + 5x + 100 dollars
Shop B: C_B = 8x + 40 dollars
For an order of 150 posters, which shop is cheaper, and by how much?
Hint
Evaluate both formulas at x = 150. Square 150 before multiplying by 0.02. Then subtract the smaller cost from the larger.
The true statements are p(-1) = 0, p(2) = p(3) and p(4) > p(3).
Check: since p(-1) = p(2) = p(3) = 0, the polynomial factors as (x + 1)(x - 2)(x - 3). Then p(4) = 5 \cdot 2 \cdot 1 = 10 ✓ and p(-2) = (-1)(-4)(-5) = -20 ✓
Tip:p(0) always equals the constant term, so you can read it without any work.
Question 36 of 36
Drop-down
In a science lab, the height in meters of a ball thrown upward is h = -4.9t^2 + 20t + 1.5, where t is the time in seconds. Sam found the height at t = 3:
Step
Work
1
-4.9(3)^2 + 20(3) + 1.5
2
-4.9(9) + 60 + 1.5
3
44.1 + 60 + 1.5
4
105.6
Sam's first error is in . The correct height at t = 3 is meters.
Choose an answer from each menu.
Hint
Check the sign of -4.9 \times 9 in Step 3.
Step-by-step solution
Step 1: substituting t = 3 is correct.
Step 2:3^2 = 9 ✓ and 20 \cdot 3 = 60 ✓. Correct.
Step 3:-4.9 \cdot 9 = -44.1 (negative times positive is negative). Sam wrote +44.1. Error.
Correct finish: -44.1 + 60 + 1.5 = 17.4.
Sam's first error is in Step 3, and the correct height is 17.4 meters.
Check: a ball thrown up at 20 m/s cannot be 105.6 m high after 3 s; at t = 2 the model gives -19.6 + 40 + 1.5 = 21.9 m, so 17.4 m at t = 3 (coming down) makes sense ✓
Tip: the leading coefficient -4.9 (or -16 in feet) is negative because gravity pulls down. That term should always reduce the height.
