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Factor the trinomial first: two numbers that multiply to -30 and add to +1.
Step-by-step solution
Look for two numbers that multiply to -30 and add to +1 (the coefficient of x).
A negative product means the numbers have opposite signs. The positive one must be larger because the sum is positive.
Try pairs: 6 and -5 give 6 \cdot (-5) = -30 and 6 + (-5) = 1 ✓.
So x^2 + x - 30 = (x + 6)(x - 5).
The factors are x + 6 and x - 5. Only x - 5 appears among the choices.
Answer: x - 5
Check:(x + 6)(x - 5) = x^2 - 5x + 6x - 30 = x^2 + x - 30 ✓. You can also test x = 5: 25 + 5 - 30 = 0. A value that makes the polynomial zero tells you x - 5 is a factor.
Tip: To test a choice like x - k, plug in x = k. If the polynomial equals 0, then x - k is a factor.
Question 2 of 10 · Write linear expressions from context
A group of 20 people visits a zoo. Adult tickets cost $12 and child tickets cost $7. If a people in the group are adults and everyone else is a child, which expression gives the total cost of the tickets, in dollars?
Hint
If a of the 20 people are adults, how many are children? Write it with a.
Step-by-step solution
Adults: a. Children are the rest of the 20 people: 20 - a.
Cost of adult tickets: 12a.
Cost of child tickets: 7(20 - a).
Total: 12a + 7(20 - a).
The expression is 12a + 7(20 - a). (Simplified, it is 5a + 140.)
Check: if the group has 5 adults and 15 children, the cost is 5(12) + 15(7) = 60 + 105 = 165 dollars. The expression gives 12(5) + 7(15) = 165 ✓
Tip: when a total is split into two groups, one group is x and the other is \text{total} - x, with the total written first.
Question 3 of 10 · Evaluate linear expressions
Drag and drop
Drag a value into each empty box to complete the table for the expression -2x + 5. Not every tile is used.
x
-3
0.5
4
-2x + 5
Tap a tile, then tap a box to place it.
Hint
The coefficient is -2. For x = -3, think: negative times negative.
Step-by-step solution
x = -3: -2(-3) + 5 = 6 + 5 =11.
x = 0.5: -2(0.5) + 5 = -1 + 5 =4.
x = 4: -2(4) + 5 = -8 + 5 =-3.
Unused tiles: -1 comes from (-2)(-3) = -6 (sign error), and 13 comes from (-2)(4) = +8 (sign error).
Check: as x goes up, -2x + 5 should go down (the coefficient is negative). The values 11, 4, -3 go down ✓. The step from x = 0.5 to x = 4 is 3.5, and 3.5 \times (-2) = -7: 4 - 7 = -3 ✓
Tip: a negative coefficient makes the values decrease as x increases. If your table goes up, look for a sign error.
Question 4 of 10 · Evaluate polynomials
Priya evaluated x^2 - 3xy + y^2 when x = -1 and y = 4. Her work is shown.
Step
Work
1
(-1)^2 - 3(-1)(4) + (4)^2
2
-1 + 12 + 16
3
27
Which statement correctly describes Priya's work?
Hint
Check each part of Step 2 separately: (-1)^2, then -3(-1)(4), then (4)^2.
Step-by-step solution
Step 1: the values are substituted correctly, each in parentheses.
Step 2: check each term.
(-1)^2 = (-1)(-1) = 1, but Priya wrote -1. Error.
-3(-1)(4): two negative factors, so +12 ✓.
(4)^2 = 16 ✓.
Correct Step 2: 1 + 12 + 16; correct value: 29.
So the correct statement is: in Step 2, (-1)^2 should be 1, not -1; the correct value is 29.
Check:x^2 - 3xy + y^2 = (x - y)^2 - xy. With x - y = -5 and xy = -4: 25 - (-4) = 29 ✓
Tip: a negative number squared is always positive. Typing (−)1 x² without parentheses on many calculators gives -1, so always use parentheses.
Question 5 of 10 · Rational expression operations
Select all
Select all expressions that are equivalent to \dfrac{10 - 2x}{x^2 - x - 20} for all values of x where both are defined.
Select all that apply.
Hint
Factor -2 out of the numerator and factor the denominator. Remember that moving a negative sign between top and bottom does not change the value.
Tip: "Work backwards" questions turn into equations. Clear the fraction by multiplying by the denominator, and remember to distribute to every term inside the parentheses (6 \cdot xand6 \cdot 1).
Question 7 of 10 · Write rational expressions from context
No calculator
Which situation can be represented by the expression \dfrac{60}{x + 3}?
Hint
Translate each situation into an expression yourself. Watch where the 3 goes: inside the denominator, outside the fraction, or in the numerator.
Step-by-step solution
A: Speed x + 3 mph over 60 miles. Time = distance ÷ speed = \dfrac{60}{x + 3}. Matches.
B: Driving 60 miles at x mph takes \dfrac{60}{x} hours; adding a 3-hour stop gives \dfrac{60}{x} + 3. Does not match.
C: A $60 bill plus a $3 tip shared by x people: \dfrac{63}{x}. Does not match.
D: x + 3 miles at 60 mph takes \dfrac{x + 3}{60} hours. Does not match (upside down).
Answer: A — the number of hours to travel 60 miles at a speed 3 mph faster than x mph
Check: Let x = 9: at 12 mph, 60 miles take 5 hours, and \dfrac{60}{9 + 3} = 5 ✓. Choice B would give \dfrac{60}{9} + 3 \approx 9.67 hours.
Tip: Where the "+3" sits tells the story. Inside the denominator, it changes the rate or group size; outside the fraction, it adds to the answer; in the numerator, it adds to the total.
Question 8 of 10 · Linear expressions: combine, factor, expand
Tip: a missing power needs a 0 placeholder in long or synthetic division. On multiple choice, multiplying each choice by (x - 2) is often just as fast.
Question 10 of 10 · Write polynomial expressions from context
Fill in
A rectangular metal plate measures (3x + 2) cm by (2x + 5) cm. Two square holes, each with sides of x cm, are cut out of the plate.
Write a simplified polynomial for the area of the remaining metal. Then find that area, in square centimeters, when x = 3.
cm²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Remaining area = plate area − 2 × (one hole's area). Simplify first, then substitute.