1Subtracting Polynomials
The ratio of a person's weight on Earth to the person's weight on the Moon is constant. An astronaut who weighs 540 newtons on Earth weighs 90 newtons on the Moon. Another astronaut weighs w newtons on Earth. Which expression represents this astronaut's weight, in newtons, on the Moon?
Answer Explanation
Find the constant ratio of Moon weight to Earth weight: 90/540 = 1/6.
Apply that same ratio to the second astronaut: Moon weight = (1/6)w = w/6. The correct answer is D.
2Expanding a Squared Binomial
Which of the following expressions is equivalent to (a/4 + b/3)²?
- A) a²/4 + b²/3
- B) a²/16 + b²/9
- C) a²/4 + ab/12 + b²/3
- D) a²/16 + ab/6 + b²/9
Answer Explanation
Use (p + q)² = p² + 2pq + q² with p = a/4 and q = b/3.
p² = a²/16, q² = b²/9, and 2pq = 2(a/4)(b/3) = ab/6. Combining: a²/16 + ab/6 + b²/9. The correct answer is D.
3Writing an Equation from a Word Problem
Jaqueline spent $150 for supplies and gas to start a lawn-mowing service. She charges $25 for each lawn she mows. In the first week Jaqueline made $50 after the cost of supplies and gas was deducted. Which equation represents this situation, where x is the number of lawns Jaqueline mowed during the first week?
- A) 25x − 150 = 50
- B) 50 − 25x = 150
- C) 150 − 25x = 50
- D) 25x + 50 = 150
Answer Explanation
Her total earnings from mowing are 25x. Subtracting the $150 startup cost gives her net profit: 25x − 150.
That net profit equaled $50: 25x − 150 = 50. The correct answer is A.
4Multiplying Polynomials
Which expression is equivalent to (3x³ − x² + 4)(5x² + 8x)?
- A) 15x⁵ + 19x⁴ − 8x³ − 20x + 32
- B) 15x⁵ + 19x⁴ − 8x³ − 20x² + 32
- C) 15x⁵ + 19x⁴ − 8x³ − 20x² + 32x
- D) 15x⁵ + 29x⁴ − 8x³ − 20x² + 32x
Answer Explanation
Distribute each term of the first polynomial across the second: 3x³(5x²+8x) = 15x⁵+24x⁴; −x²(5x²+8x) = −5x⁴−8x³; 4(5x²+8x) = 20x²+32x.
Combine like terms: 15x⁵ + (24x⁴−5x⁴) − 8x³ + 20x² + 32x = 15x⁵+19x⁴−8x³+20x²+32x. The correct answer is C.
5Solving a Linear Equation
If (6/7)p + 12 = 54, what is the value of 7p?
Answer Explanation
Subtract 12 from both sides: (6/7)p = 42. Multiply both sides by 7/6: p = 49.
Then 7p = 7(49) = 343.
6Solving a Literal Equation
b² + 5c = 9d. The given equation relates the real numbers b, c, and d, where d > (5/9)c. Which equation correctly expresses b in terms of c and d?
- A) b = (9d + 5c)/2
- B) b = (9d − 5c)/2
- C) b = ±√(9d + 5c)
- D) b = ±√(9d − 5c)
Answer Explanation
Isolate b² by subtracting 5c from both sides: b² = 9d − 5c.
Take the square root of both sides, keeping both roots since b² = k has two solutions: b = ±√(9d − 5c). The condition d > (5/9)c just guarantees the expression under the root is positive. The correct answer is D.
7Using a Difference of Cubes Pattern
If x³ + y³ = 12 and x³ − y³ = 6, what is the value of x⁶ − y⁶?
Answer Explanation
Recognize that x⁶ − y⁶ factors as a difference of squares: (x³)² − (y³)² = (x³ + y³)(x³ − y³).
Substitute the given values directly: (12)(6) = 72.
8Combining Like Terms
Which expression is equivalent to 2(xy² + x²y) − 8(xy² − 4x²y)?
- A) −6xy² + 34x²y
- B) −6xy² + 33x²y
- C) −6xy² − 30x²y
- D) −6xy² − 31x²y
Answer Explanation
Distribute: 2(xy²+x²y) = 2xy²+2x²y, and −8(xy²−4x²y) = −8xy²+32x²y.
Combine like terms: (2xy²−8xy²) + (2x²y+32x²y) = −6xy²+34x²y. The correct answer is A.
9Solving a Linear Equation
If 2x + 3 = 9, what is the value of 4x − 4?
Answer Explanation
Solve for x first: 2x = 6, so x = 3.
Substitute into the target expression: 4(3) − 4 = 12 − 4 = 8.
10Rewriting a Radical Expression
Which expression is equivalent to (131y)1/2, where y > 1?
- A) 131√y
- B) √131y
- C) √(131y)
- D) √(131y)²
Answer Explanation
A power of 1/2 is equivalent to a square root: (131y)1/2 = √(131y).
This keeps the entire product 131y under the radical, rather than pulling 131 outside. The correct answer is C.
11Subtracting Polynomials
Which expression is equivalent to (5a² − 3ab + 7b²) − (4a² + ab − 2b²)?
- A) a² − 4ab + 9b²
- B) a² − 2ab + 5b²
- C) a⁴ − 4a²b² + 9b⁴
- D) a⁴ − 2a²b² + 5b⁴
Answer Explanation
Distribute the negative sign across the second polynomial: 5a²−3ab+7b²−4a²−ab+2b².
Combine like terms: (5a²−4a²) + (−3ab−ab) + (7b²+2b²) = a²−4ab+9b². The correct answer is A.
12Solving a Literal Equation
7x + 16 = 4/y. The given equation relates the distinct positive numbers x and y. Which equation correctly expresses y in terms of x?
- A) y = 7x + 12
- B) y = 4/(7x + 16)
- C) y = (7x + 16)/4
- D) y = 4(7x + 16)
Answer Explanation
Starting from 7x + 16 = 4/y, multiply both sides by y: y(7x + 16) = 4.
Divide both sides by (7x + 16): y = 4/(7x + 16). The correct answer is B.
13Writing an Equation from a Word Problem
A total of 6 squares each have side length r. A total of 3 equilateral triangles each have side length t. None of these squares and triangles shares a side. The sum of the perimeters of all these squares and triangles is 75. Which equation represents this situation?
- A) 18r + 12t = 75
- B) 24r + 9t = 75
- C) 3r + 6t = 75
- D) 6r + 3t = 75
Answer Explanation
Each square has 4 sides of length r, so 6 squares contribute a total perimeter of 6(4r) = 24r. Each triangle has 3 sides of length t, so 3 triangles contribute 3(3t) = 9t.
The combined perimeter is 24r + 9t = 75. The correct answer is B.
14Solving a Literal Equation
v = −w/(130x). The given equation relates the distinct positive numbers v, w, and x. Which equation correctly expresses w in terms of v and x?
- A) w = −130vx
- B) w = −130v/x
- C) w = −x/(130v)
- D) w = v + 130x
Answer Explanation
Multiply both sides of v = −w/(130x) by 130x to isolate w: 130xv = −w.
Multiply both sides by −1: w = −130vx. The correct answer is A.
15Distributing a Constant
Which expression is equivalent to 15(x + 10)?
- A) 15x + 10
- B) 15x + 150
- C) 15x + 25
- D) 15x + 5
Answer Explanation
Distribute the 15 across both terms inside the parentheses: 15(x) + 15(10).
That gives 15x + 150. The correct answer is B.
16Combining Like Terms
Which expression is equivalent to 60y + 90y?
- A) 540y²
- B) 540y
- C) 150y²
- D) 150y
Answer Explanation
Both terms share the variable y to the first power, so they combine directly by adding coefficients: 60 + 90 = 150.
That gives 150y (not 150y², since no multiplication of variables occurs here). The correct answer is D.
17Solving a Linear Equation
If 9(x + 3) = 81, what is the value of x + 3?
Answer Explanation
Treat (x + 3) as a single unit. Divide both sides of 9(x + 3) = 81 by 9.
That gives x + 3 = 9 directly — no need to solve for x itself. The correct answer is B.
18Combining Like Terms
Which expression is equivalent to (9nr² + 3nr) + (6n²r + 5nr)?
- A) 15n²r² + 8nr
- B) 15n³r³ + 8n²r²
- C) 54n³r³ + 15n²r²
- D) 9nr² + 6n²r + 8nr
Answer Explanation
Remove the parentheses (addition doesn't change any signs): 9nr² + 3nr + 6n²r + 5nr.
Combine only the truly like terms — 3nr and 5nr share the exact same variables and exponents, but 9nr² and 6n²r do not match each other or the nr terms: 9nr² + 6n²r + 8nr. The correct answer is D.
19Solving a Literal Equation
v = −w/(161x). The given equation relates the distinct positive numbers v, w, and x. Which equation correctly expresses w in terms of v and x?
- A) w = −161vx
- B) w = −161v/x
- C) w = −x/(161v)
- D) w = v + 161x
Answer Explanation
Multiply both sides of v = −w/(161x) by 161x to isolate w: 161xv = −w.
Multiply both sides by −1: w = −161vx. The correct answer is A.
20Distributing a Constant
Which expression is equivalent to 16(x + 5)?
- A) 16x + 5
- B) 16x + 80
- C) 16x + 21
- D) 16x + 11
Answer Explanation
Distribute the 16 across both terms inside the parentheses: 16(x) + 16(5).
That gives 16x + 80. The correct answer is B.
21Solving a Literal Equation
x = 2a(b + 5). The given equation relates the positive numbers a, b, and x. Which equation correctly expresses a in terms of b and x?
- A) a = x/2 − (b + 5)
- B) a = x/(2(b + 5))
- C) a = 2(b + 5)/x
- D) a = 2x(b + 5)
Answer Explanation
Divide both sides of x = 2a(b + 5) by 2(b + 5) to isolate a.
That gives a = x/(2(b + 5)). The correct answer is B.
22Simplifying a Rational Expression
Which expression is equivalent to (y + 2)/(x − 9) + y(x − 9)/(x²y − 9xy)?
- A) (xy + y − 7)/(x³y − 18x²y + 81xy)
- B) (xy + 10y + 2)/(x²y − 9xy + x − 9)
- C) (xy² + 3xy − 9y)/(x²y − 9xy)
- D) (xy² + 3xy − 9y)/(x³y − 18x²y + 81xy)
Answer Explanation
Factor the second denominator: x²y − 9xy = xy(x − 9), which is already the common denominator needed.
Rewrite the first fraction over that same denominator: (y+2)/(x−9) = xy(y+2)/[xy(x−9)] = (xy²+2xy)/(x²y−9xy). Adding the second fraction's numerator, y(x−9) = xy−9y: (xy²+2xy) + (xy−9y) = xy²+3xy−9y. The correct answer is C.
23Solving for a Constant
If x/y = 63 and cx/(3y) = 63, what is the value of c?
Answer Explanation
Rewrite cx/(3y) as (c/3)(x/y). Since x/y = 63, this becomes (c/3)(63) = 21c.
Set that equal to 63: 21c = 63, so c = 3.
24Solving a Literal Equation
x = 2a(b + 3). The given equation relates positive numbers x, a, and b. Which equation correctly expresses a in terms of b and x?
- A) a = x/2 − 4b + 3
- B) a = x/(2(b + 3))
- C) a = 2(b + 3)/x
- D) a = 2x(b + 3)
Answer Explanation
Divide both sides of x = 2a(b + 3) by 2(b + 3) to isolate a.
That gives a = x/(2(b + 3)). The correct answer is B.
25Solving a Multi-Step Linear Equation
If 8 + 9(6 − x)/4 = 7(6 − x)/3, what is the value of 6 − x?
Answer Explanation
Let u = 6 − x to simplify. The equation becomes 8 + 9u/4 = 7u/3. Multiply every term by 12 (the LCD of 4 and 3): 96 + 27u = 28u.
Subtract 27u from both sides: 96 = u. So 6 − x = 96.
26Solving a Linear Equation
If (7/8)p + 35 = 56, what is the value of 8p?
Answer Explanation
Subtract 35 from both sides: (7/8)p = 21. Multiply both sides by 8/7: p = 24.
Then 8p = 8(24) = 192.
27Writing a Function from a Description
The function f gives the product of a number, x, and a number that is 66 more than x. Which equation defines f?
- A) f(x) = x² + x + 66
- B) f(x) = x² + 66
- C) f(x) = x² + 66x
- D) f(x) = x² + 66x + 66
Answer Explanation
"A number that is 66 more than x" translates to x + 66. The function is the product of x and that expression: f(x) = x(x + 66).
Distribute: f(x) = x² + 66x. The correct answer is C.
28Subtracting Polynomials
(5x³ − 2) − (−6x³ + 3). The given expression is equivalent to bx³ − 5, where b is a constant. What is the value of b?
Answer Explanation
Distribute the negative sign: 5x³ − 2 + 6x³ − 3.
Combine like terms: (5x³+6x³) + (−2−3) = 11x³ − 5. Comparing to bx³ − 5 gives b = 11.
29Subtracting Polynomials
Which expression is equivalent to (4x² + 6) − (3x²)?
- A) 7
- B) 7x² + 6
- C) x² − 6
- D) x² + 6
Answer Explanation
Distribute the subtraction: 4x² + 6 − 3x².
Combine like terms: (4x²−3x²) + 6 = x² + 6. The correct answer is D.
30Adding Polynomials
Which expression is equivalent to (x³ + 6x² − 8x) + 8(x² + 4)?
- A) x³ + 4x² − 8x + 32
- B) x³ + 14x² − 8x + 32
- C) x³ + 8x² − 8x + 32
- D) x³ + 6x² − 8x + 32
Answer Explanation
Distribute the 8: 8(x²+4) = 8x²+32. Then add to the first polynomial: x³+6x²−8x+8x²+32.
Combine like terms: x³ + (6x²+8x²) − 8x + 32 = x³+14x²−8x+32. The correct answer is B.
31Solving a Quadratic Word Problem
The product of a positive number x and the number that is 1 less than x is equal to 342. What is the value of x?
Answer Explanation
Translate directly: x(x − 1) = 342, which expands to x² − x − 342 = 0.
Factor or use the quadratic formula: x = (1 ± √(1 + 1368))/2 = (1 ± 37)/2, giving x = 19 or x = −18. Since x is positive, x = 19. The correct answer is B.
32Simplifying a Rational Expression
Which expression is equivalent to (y + 4)/(x − 7) + y(x − 7)/(x²y − 7xy)?
- A) (xy + y − 3)/(x³y − 14x²y + 49xy)
- B) (xy + 8y + 4)/(x²y − 7x²y + x − 7)
- C) (xy² + 5xy − 7y)/(x²y − 7xy)
- D) (xy² + 5xy − 7y)/(x³y − 14x²y + 49xy)
Answer Explanation
Factor the second denominator: x²y − 7xy = xy(x − 7), which is the common denominator to use.
Rewrite the first fraction over that denominator: (y+4)/(x−7) = xy(y+4)/[xy(x−7)] = (xy²+4xy)/(x²y−7xy). The second fraction's numerator is y(x−7) = xy−7y. Adding: (xy²+4xy)+(xy−7y) = xy²+5xy−7y. The correct answer is C.
33Solving for a Constant Ratio
If x/y = 3 and 5x/(ty) = 135, what is the value of t?
Answer Explanation
Rewrite 5x/(ty) as (5/t)(x/y). Since x/y = 3, this becomes (5/t)(3) = 15/t.
Set that equal to 135: 15/t = 135, so t = 15/135 = 1/9. The correct answer is A.
34Writing a System from a Word Problem
A piece of string with a length of 35 inches is cut into two parts. One part has a length of x inches, and the other part has a length of y inches. The value of x is 5 more than 2 times the value of y. What is the value of x?
Answer Explanation
The two parts add up to the total length: x + y = 35. The second sentence translates to x = 2y + 5.
Substitute into the first equation: (2y + 5) + y = 35, so 3y + 5 = 35, giving 3y = 30 and y = 10. Then x = 2(10) + 5 = 25.
35Adding Polynomials
Which expression is equivalent to (x³ + 4x² − 3x) + 5(x² + 8)?
- A) x³ + 9x² − 3x + 40
- B) x³ + 4x² − 3x + 40
- C) x³ + 8x² − 3x + 40
- D) x³ + 5x² − 3x + 40
Answer Explanation
Distribute the 5: 5(x²+8) = 5x²+40. Add to the first polynomial: x³+4x²−3x+5x²+40.
Combine like terms: x³ + (4x²+5x²) − 3x + 40 = x³+9x²−3x+40. The correct answer is A.
36Solving a Literal Equation
2/c − 7/d = 5. The given equation relates the variables c and d, where c and d are greater than 1. Which equation correctly expresses d in terms of c?
- A) d = c/(7(2 − 5c))
- B) d = 7c/(2 − 5c)
- C) d = (7(2 − 5c))/c
- D) d = (2 − 5c)/(7c)
Answer Explanation
Isolate the d-term: −7/d = 5 − 2/c. Combine the right side over a common denominator: 5 − 2/c = (5c − 2)/c, so −7/d = (5c − 2)/c, or equivalently 7/d = (2 − 5c)/c.
Take the reciprocal of both sides: d/7 = c/(2 − 5c), so d = 7c/(2 − 5c). The correct answer is B.
