Factoring — Practice Set

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

Factoring means rewriting an expression as a product: pull out the common factor first, then match a pattern, because only a product tells you where something is zero or what will cancel.

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-FAC-001 · MCQ · 45s

Which of the following shows 12x^3 + 18x^2 factored completely over the integers?

(A) 6x(2x^2 + 3x)
(B) 6x^2(2x + 3)
(C) 3x^2(4x + 6)
(D) 2x^2(6x + 9)
(E) 12x^2(x + 1.5)

Q2 · ALG-FAC-012 · MCQ · 45s

Factor: x^2 - 14x + 49.

(A) (x - 7)^2
(B) (x + 7)^2
(C) (x - 7)(x + 7)
(D) (x - 49)(x - 1)
(E) (x - 49)(x + 1)

Q3 · ALG-FAC-006 · QC · 50s

x^2 - 16 = 0, where x is a positive integer.

Column A: x Column B: 4

Q4 · ALG-FAC-011 · Numeric Entry · 50s

What is the value of sqrt(85^2 - 84^2)? Enter your answer as a number.

Numeric entry — write the number.

Q5 · ALG-FAC-013 · Select all that apply · 60s

Which of the following expressions are equal to (x^2 - 25)/(x^2 + 10x + 25) for every real x with x not equal to -5? Select all that apply.

(A) (x - 5)/(x + 5)
(B) (x + 5)/(x - 5)
(C) 1 - 10/(x + 5)
(D) 1 - 5/(x + 5)
(E) -25/(10x + 25)

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · ALG-FAC-014 · MCQ · 80s

Factor completely: 6x^2 - 5x - 6.

(A) (3x + 2)(2x - 3)
(B) (3x - 2)(2x + 3)
(C) (6x + 1)(x - 6)
(D) (6x + 2)(x - 3)
(E) (3x + 6)(2x - 1)

Q7 · ALG-FAC-026 · MCQ · 75s

For x not equal to -3, which of the following is equal to (x^2 - x - 12)/(x + 3)?

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

Q8 · ALG-FAC-032 · MCQ · 70s

Which of the following is equivalent to (a + b)^2 - (a - b)^2?

(A) 2b^2
(B) 4a^2
(C) 4ab
(D) 2ab
(E) 4a^2b^2

Q9 · ALG-FAC-017 · QC · 80s

a and b are real numbers with a > b > 0.

Column A: (a - b)^2 Column B: a^2 - b^2

Q10 · ALG-FAC-021 · QC · 75s

x is a real number and x^3 = 9x.

Column A: the number of distinct real values of x that satisfy the equation Column B: 3

Q11 · ALG-FAC-019 · Numeric Entry · 75s

(x + p)(x + q) = x^2 + 3x - 28 for all x, where p and q are integers. What is the value of |p - q|? Enter your answer as a number.

Numeric entry — write the number.

Q12 · ALG-FAC-030 · Numeric Entry · 80s

The solutions of (x - 3)(x + 5) = 9 are x = p and x = q, where p > q. What is the value of p - q? Enter your answer as a number.

Numeric entry — write the number.

Q13 · ALG-FAC-020 · Select all that apply · 85s

Which of the following expressions are equal to 0 when x = 4? Select all that apply.

(A) x^2 - 8x + 16
(B) x^2 + 16
(C) x^2 - 2x - 8
(D) x^2 - 4x - 32
(E) x^3 - 16x

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Level 3 · Hard — 12 questions · ~21 min

where 162+ is won or lost

Q14 · ALG-FAC-034 · MCQ · 110s

For x not equal to 2, which of the following is equal to [(x^4 - 16)/(x^3 - 8)] * [(x^2 + 2x + 4)/(x^2 + 4)]?

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

Q15 · ALG-FAC-035 · MCQ · 110s

Factor completely: 4x^4 - 37x^2 + 9.

(A) (2x - 1)(2x + 1)(x - 3)(x + 3)
(B) (2x - 1)^2(x - 3)^2
(C) (2x - 3)(2x + 3)(x - 1)(x + 1)
(D) (2x - 1)(2x + 1)(x^2 + 9)
(E) 4(x^2 - 1)(x^2 - 9)

Q16 · ALG-FAC-036 · Numeric Entry · 100s

The expression x^3 + x^2 - 4x - 4 factors completely as (x + a)(x + b)(x + c), where a, b and c are integers. What is the value of a + b + c? Enter your answer as a number.

Numeric entry — write the number.

Q17 · ALG-FAC-037 · MCQ · 110s

Which of the following is the complete factorisation of x^8 - 256 into factors that cannot be factored further over the integers?

(A) (x - 2)(x + 2)(x^2 + 4)(x^4 + 16)
(B) (x^2 - 4)(x^2 + 4)(x^4 + 16)
(C) (x - 2)^2(x + 2)^2(x^4 + 16)
(D) (x - 2)(x + 2)(x^2 + 4)^3
(E) (x - 2)(x + 2)(x^2 + 4)(x^4 - 16)

Q18 · ALG-FAC-038 · QC · 105s

x is a nonzero real number.

Column A: x^2 + 1/x^2 Column B: 2

Q19 · ALG-FAC-039 · Select all that apply · 110s

x and y are positive integers with x > y and x^2 - y^2 = 84. Which of the following could be the value of x + y? Select all that apply.

(A) 14
(B) 21
(C) 28
(D) 42
(E) 84

Q20 · ALG-FAC-040 · MCQ · 95s

If x^2 - y^2 = 60 and x - y = 5, what is the value of x^2 + 2xy + y^2?

(A) 12
(B) 25
(C) 60
(D) 144
(E) 3600

Q21 · ALG-FAC-041 · Numeric Entry · 110s

x, y and z are real numbers with x + y + z = 9, x^2 + y^2 + z^2 = 41 and xyz = 12. What is the value of (x + y)(y + z)(z + x)? Enter your answer as a number.

Numeric entry — write the number.

Q22 · ALG-FAC-042 · MCQ · 100s

Factor completely: x^2 + 6x + 9 - 4y^2.

(A) (x + 3 - 2y)(x + 3 + 2y)
(B) (x + 3 - 4y)(x + 3 + 4y)
(C) (x + 3 - 2y)^2
(D) (x + 9 - 2y)(x + 9 + 2y)
(E) (x + 3)(x + 3 - 4y^2)

Q23 · ALG-FAC-043 · QC · 105s

k is a positive integer, and x^2 - kx + 12 can be factored into two binomials with integer coefficients.

Column A: the number of possible values of k Column B: 4

Q24 · ALG-FAC-044 · Select all that apply · 105s

x^3 - 7x + 6 = (x - 1)(x - a)(x - b) for all x, where a and b are integers with a > b. Which of the following are true? Select all that apply.

(A) a + b = -1
(B) ab = -6
(C) a - b = 5
(D) The equation x^3 - 7x + 6 = 0 has three positive solutions
(E) a = 2

Q25 · ALG-FAC-045 · Numeric Entry · 110s

What is the greatest prime factor of 2^18 - 1? Enter your answer as a number.

Numeric entry — write the number.

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Level 4 · Extreme — 5 questions · ~11 min

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

Q26 · ALG-FAC-046 · MCQ · 125s

p and q are real numbers with p - q = 2 and p^3 - q^3 = 98. What is the value of p^2 + q^2?

(A) 4
(B) 34
(C) 45
(D) 49
(E) 64

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

x^4 + 4x^2 + 16 = (x^2 + ax + 4)(x^2 + bx + 4) for all x, where a and b are integers. What is the value of a^2 + b^2? Enter your answer as a number.

Numeric entry — write the number.

Q28 · ALG-FAC-048 · QC · 130s

x^4 - 10x^3 + 33x^2 - 40x + 16 = 0

Column A: the product of all DISTINCT real solutions of the equation Column B: the sum of all DISTINCT real solutions of the equation

Q29 · ALG-FAC-049 · Select all that apply · 125s

Which of the following are factors of x^6 - 1? Select all that apply.

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

Q30 · ALG-FAC-050 · MCQ · 130s

x and y are integers with xy + 5x - 3y = 27. How many ordered pairs (x, y) satisfy this equation?

(A) 3
(B) 6
(C) 12
(D) 24
(E) There are infinitely many.

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