Algebraic Expressions — Practice Set

Bank code: ALG-EXP · Section: Algebra · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

An expression is simplified, not solved: rewrite it with the identities you know cold, cancel factors but never terms, and the real question usually collapses in one move.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · ALG-EXP-001 · MCQ · 40s

Simplify: 3(x + 4).

(A) 3x + 4
(B) 3x + 7
(C) 3x + 12
(D) x + 12
(E) 7x

Q2 · ALG-EXP-005 · MCQ · 45s

If x is not equal to 0, simplify: (x^2 + x) / x.

(A) x^2 + 1
(B) x + 1
(C) x^2
(D) x
(E) 1 + 1/x

Q3 · ALG-EXP-008 · QC · 50s

x is a positive integer.

Column A: (x + 2)^2 Column B: x^2 + 4

Q4 · ALG-EXP-010 · Numeric Entry · 55s

Simplify 4(2x^2 - 1) - 3(x^2 + 2) and evaluate the result at x = 3. Enter your answer as a number.

Numeric entry — write the number.

Q5 · ALG-EXP-012 · Select all that apply · 60s

Which of the following expressions are equivalent to x^2 - 16 for every real x? Select all that apply.

(A) (x - 4)(x + 4)
(B) (x - 4)^2
(C) (x - 2)(x + 2) - 12
(D) (x - 8)(x + 2)
(E) x^2 - 4^2 - 8x

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · ALG-EXP-014 · MCQ · 65s

Expand and simplify: (2x + 3)(3x - 4).

(A) 6x^2 - 12
(B) 6x^2 + x - 12
(C) 6x^2 - x - 12
(D) 6x^2 + 17x - 12
(E) 6x^2 - 17x - 12

Q7 · ALG-EXP-020 · MCQ · 80s

If x and y are nonzero, simplify: (1/x + 1/y) / (1/(xy)).

(A) x + y
(B) xy(x + y)
(C) 1/(x + y)
(D) (x + y)/(xy)
(E) xy

Q8 · ALG-EXP-022 · Numeric Entry · 85s

If x - y = 4 and x + y = 10, what is the value of x^4 - y^4? Enter your answer as a number.

Numeric entry — write the number.

Q9 · ALG-EXP-025 · Select all that apply · 75s

For which of the following values of x is the expression (x^2 - x - 6) / (x^2 - 9) undefined? Select all that apply.

(A) x = -3
(B) x = -2
(C) x = 0
(D) x = 2
(E) x = 3

Q10 · ALG-EXP-026 · QC · 75s

p and q are positive numbers with p > q.

Column A: (p + q)^2 Column B: (p - q)^2 + 4pq

Q11 · ALG-EXP-027 · Numeric Entry · 65s

If a^2 + b^2 = 25 and ab = 12, what is the value of (a + b)^2? Enter your answer as a number.

Numeric entry — write the number.

Q12 · ALG-EXP-029 · MCQ · 80s

Expand: (x + y + z)^2.

(A) x^2 + y^2 + z^2
(B) x^2 + y^2 + z^2 + 2xy + 2yz + 2xz
(C) x^2 + y^2 + z^2 + xy + yz + xz
(D) x^2 + y^2 + z^2 + 2xyz
(E) x^2 + y^2 + 2xy + z^2

Q13 · ALG-EXP-051 · MCQ · 80s

At a bookshop a hardcover costs h dollars, and a paperback costs 4 dollars less than a hardcover. Jo buys 3 hardcovers and 5 paperbacks. Which of the following expresses Jo's total cost, in dollars?

(A) 8h - 20
(B) 8h - 4
(C) 8h + 20
(D) 15h - 20
(E) 8h - 32

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Level 3 · Hard — 12 questions · ~21 min

where 162+ is won or lost

Q14 · ALG-EXP-034 · MCQ · 110s

For all real x, (8x^3 - 27)/(4x^2 + 6x + 9) - (8x^3 + 27)/(4x^2 - 6x + 9) is equal to which of the following?

(A) -6
(B) 0
(C) 6
(D) 4x
(E) 4x - 6

Q15 · ALG-EXP-035 · Numeric Entry · 105s

If x + y = 6 and x^2 + y^2 = 20, what is the value of x^3 + y^3? Enter your answer as a number.

Numeric entry — write the number.

Q16 · ALG-EXP-036 · QC · 95s

x and y are real numbers with x^2 + y^2 = 10 and xy = -3.

Column A: (x + y)^2 Column B: (x - y)^2

Q17 · ALG-EXP-037 · MCQ · 110s

For all x, 2x^3 - 3x^2 - 5x - 12 = (x + r)(2x^2 + bx + c), where r, b and c are integers. What is the value of b + c?

(A) -13
(B) -9
(C) -5
(D) 1
(E) 7

Q18 · ALG-EXP-038 · Select all that apply · 100s

k is a constant and 4x^2 + kx + 25 is the square of a binomial with integer coefficients. Which of the following could be the value of k? Select all that apply.

(A) -20
(B) -10
(C) 10
(D) 20
(E) 40

Q19 · ALG-EXP-039 · MCQ · 110s

x is a real number with x not equal to 1, 2 or 3. Which of the following is equivalent to 1/(x^2 - 3x + 2) - 1/(x^2 - 4x + 3)?

(A) -1/[(x - 1)(x - 2)(x - 3)]
(B) 1/[(x - 1)(x - 2)(x - 3)]
(C) -1/[(x - 2)(x - 3)]
(D) (2x - 5)/[(x - 1)(x - 2)(x - 3)]
(E) -5/[(x - 1)(x - 2)(x - 3)]

Q20 · ALG-EXP-040 · QC · 105s

a and b are real numbers with a > b, a + b = 7 and ab = 10.

Column A: a^3 - b^3 Column B: 3(a^2 + ab + b^2)

Q21 · ALG-EXP-041 · Numeric Entry · 100s

Simplify [(x + h)^3 - x^3] / h, where h is not equal to 0, and evaluate the result at x = 2 and h = 1. Enter your answer as a number.

Numeric entry — write the number.

Q22 · ALG-EXP-042 · MCQ · 95s

If x^2 - 5x + 6 = 0, what is the value of x^4 - 5x^3 + 6x^2 + 2?

(A) 0
(B) 2
(C) 6
(D) 11
(E) Cannot be determined from the information given

Q23 · ALG-EXP-043 · Select all that apply · 110s

a and b are real numbers with a + b = 6 and ab = 5. Which of the following expressions have the value 216? Select all that apply.

(A) a^3 + b^3 + 3ab(a + b)
(B) a^3 + b^3
(C) (a + b)(a^2 - 2ab + b^2)
(D) (a + b)(a^2 + ab + b^2)
(E) 6(a + b)^2

Q24 · ALG-EXP-044 · Numeric Entry · 105s

x is a real number with x^3 + 2x^2 = 8x. What is the sum of all possible values of x^2 + 3x + 1? Enter your answer as a number.

Numeric entry — write the number.

Q25 · ALG-EXP-045 · QC · 100s

x and y are real numbers with x > y > 0.

Column A: (x^3 - y^3) / (x - y) Column B: (x + y)^2

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Level 4 · Extreme — 5 questions · ~11 min

165+ — expect to need the insight, not the grind

Q26 · ALG-EXP-046 · MCQ · 130s

x is a positive real number with x^2 + 1/x^2 = 7. What is the value of x^5 + 1/x^5?

(A) 47
(B) 123
(C) 126
(D) 129
(E) 186

Q27 · ALG-EXP-047 · Numeric Entry · 130s

x and y are real numbers with x > y > 0, x^4 + y^4 = 2417 and xy = 14. What is the value of (x + y)^3 - (x - y)^3? Enter your answer as a number.

Numeric entry — write the number.

Q28 · ALG-EXP-048 · QC · 125s

a, b and c are real numbers with a + b + c = 0.

Column A: a^3 + b^3 + c^3 Column B: 3abc

Q29 · ALG-EXP-049 · Select all that apply · 130s

t is a real number with t^2 - 4t + 1 = 0. Which of the following expressions have the value 52? Select all that apply.

(A) t^3 + 1/t^3
(B) (t + 1/t)^3
(C) (t + 1/t)(t^2 + 1/t^2) - 4
(D) t^4 + 1/t^4
(E) (t^2 + 1/t^2)(t + 1/t)

Q30 · ALG-EXP-050 · MCQ · 130s

For all x and y with x + 2y not equal to 0, (x - 2y)(x^2 + 4y^2)(x^4 + 16y^4) is equivalent to which of the following?

(A) (x^8 - 256y^8)/(x + 2y)
(B) x^8 - 256y^8
(C) (x^8 - 256y^8)/(x - 2y)
(D) (x^8 - 16y^8)/(x + 2y)
(E) x^7 - 128y^7

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