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Question 1
Algebra IIFoundation

Algebra II practice question 1

Algebra II practice topics

Question sets mix direct skills with unfamiliar applications so method selection becomes part of the practice.

01

Polynomial and rational functions

Factor to find zeros, and tell a hole apart from a vertical asymptote when a rational function is undefined.

02

Radicals and complex numbers

Simplify radicals using perfect-square factors, and add, multiply, and rationalize expressions containing i.

03

Exponential and logarithmic models

Set up growth and decay models, and use log rules to solve equations where the unknown sits in the exponent.

04

Sequences and series

Find the nth term of arithmetic and geometric sequences and total a finite series with the sum formula.

Algebra II practice questions with worked answers

These 8 questions are printed in full on this page, with every step of the arithmetic written out. Cover the options, solve the question on paper first, and only then open the worked answer to compare your method with the one shown.

Question 1: What are all the zeros of f(x) = x^3 - 4x?

Foundation level · polynomial and rational functions

  1. x = 0, 2, -2
  2. x = 0, 4
  3. x = 2, -2
  4. x = 0, 2
Show the worked answer
  1. Every term has an x, so factor it out: x^3 - 4x = x(x^2 - 4).
  2. x^2 - 4 is a difference of two squares: x^2 - 4 = (x - 2)(x + 2).
  3. So f(x) = x(x - 2)(x + 2), and any factor equal to zero gives a zero of f.
  4. That gives x = 0, x = 2, and x = -2. Check x = -2: (-8) - (-8) = 0.

Answer: x = 0, 2, -2

Question 2: The function f(x) = (x^2 - 9)/(x - 3) is not defined at x = 3. What does its graph do there?

Exam level · polynomial and rational functions

  1. It has a hole at (3, 6)
  2. It has a vertical asymptote at x = 3
  3. It has a zero at x = 3
  4. It has a horizontal asymptote at y = 3
Show the worked answer
  1. Factor the top: x^2 - 9 = (x - 3)(x + 3).
  2. The factor (x - 3) appears on the top and the bottom, so it cancels.
  3. For every x except 3, the function is just x + 3, a straight line.
  4. At x = 3 the line would give 3 + 3 = 6, so the graph has a hole at (3, 6).

Answer: It has a hole at (3, 6)

Question 3: Simplify the square root of 72.

Foundation level · radicals and complex numbers

  1. 6 times the square root of 2
  2. 8 times the square root of 3
  3. 2 times the square root of 18
  4. 36 times the square root of 2
Show the worked answer
  1. Look for the largest perfect square that divides 72.
  2. 72 = 36 x 2, and 36 is a perfect square.
  3. The square root of a product splits: root(36 x 2) = root(36) x root(2).
  4. root(36) = 6, so the answer is 6 root(2). Check: 6^2 x 2 = 36 x 2 = 72.

Answer: 6 times the square root of 2

Question 4: Multiply the complex numbers (3 + 2i)(4 - 5i).

Exam level · radicals and complex numbers

  1. 22 - 7i
  2. 2 - 7i
  3. 12 + 10i
  4. 22 + 7i
Show the worked answer
  1. Expand every pair: 3 x 4 = 12, 3 x (-5i) = -15i, 2i x 4 = 8i, 2i x (-5i) = -10i^2.
  2. Because i^2 = -1, the term -10i^2 becomes +10.
  3. Real parts: 12 + 10 = 22.
  4. Imaginary parts: -15i + 8i = -7i, so the product is 22 - 7i.

Answer: 22 - 7i

Question 5: A colony of 400 bacteria doubles every 3 hours. How many bacteria are there after 12 hours?

Exam level · exponential and logarithmic models

  1. 6400
  2. 1600
  3. 3200
  4. 4800
Show the worked answer
  1. Count the doublings: 12 hours / 3 hours per doubling = 4 doublings.
  2. Each doubling multiplies by 2, so the total factor is 2^4 = 16.
  3. Multiply the starting amount: 400 x 16.
  4. 400 x 16 = 6400 bacteria.

Answer: 6400

Question 6: Solve log base 2 of x plus log base 2 of (x - 6) equals 4.

Challenge level · exponential and logarithmic models

  1. x = 8
  2. x = 8 or x = -2
  3. x = -2
  4. x = 10
Show the worked answer
  1. Adding two logs with the same base multiplies the insides: log2(x(x - 6)) = 4.
  2. Rewrite the log as a power: x(x - 6) = 2^4 = 16.
  3. Expand and set to zero: x^2 - 6x - 16 = 0, which factors as (x - 8)(x + 2) = 0.
  4. x = 8 or x = -2, but a log needs a positive input, so x = -2 is rejected and x = 8.

Answer: x = 8

Question 7: An arithmetic sequence starts 7, 11, 15, 19, ... What is the 20th term?

Foundation level · sequences and series

  1. 83
  2. 80
  3. 87
  4. 79
Show the worked answer
  1. The common difference is 11 - 7 = 4, and it repeats: 15 - 11 = 4.
  2. The nth term is first term + (n - 1) x difference.
  3. For n = 20: 7 + 19 x 4.
  4. 19 x 4 = 76, and 7 + 76 = 83.

Answer: 83

Question 8: Find the sum of the geometric series 3 + 6 + 12 + ... + 384.

Challenge level · sequences and series

  1. 765
  2. 768
  3. 384
  4. 1152
Show the worked answer
  1. Each term doubles, so the common ratio is 2 and the first term is 3.
  2. Find how many terms: 384 / 3 = 128 = 2^7, so the last term is the 8th term.
  3. The sum of n terms is first x (r^n - 1)/(r - 1) = 3 x (2^8 - 1)/(2 - 1).
  4. 2^8 = 256, so the sum is 3 x 255 = 765.

Answer: 765

How the three Algebra II difficulty levels differ

The difficulty buttons above change what a question asks of you, not just the size of the numbers. Each example below is taken from the question set on this page.

Difficulty levels for Algebra II practice, with an example question from this page.
LevelWhat it testsExample questionTime target
FoundationOne skill at a time, with the numbers kept small enough to check in your head.What are all the zeros of f(x) = x^3 - 4x?About 1 minute
ExamThe wording of a real test paper: pick the method first, then carry out two or three steps.The function f(x) = (x^2 - 9)/(x - 3) is not defined at x = 3. What does its graph do there?2 to 3 minutes
ChallengeTwo ideas combined, or a result you have to interpret after the calculation ends.Solve log base 2 of x plus log base 2 of (x - 6) equals 4.4 to 5 minutes

How to practice Algebra II effectively

Begin without notes and explain your choice before checking. For every miss, identify whether the cause was a definition, setup, calculation, interpretation, or time decision. Re-solve the question from a blank page, then return to the same skill in a mixed set tomorrow.

What your Algebra II answer review should show

A useful review shows more than the correct option. Compare the method with your first attempt, locate the earliest incorrect decision, and write one rule that would prevent the same error in a new Algebra II problem.

Move from mixed Algebra II questions to a complete course

This page targets flexible Algebra II question practice. When you need a syllabus-aligned sequence with unit selection, use Algebra II practice by unit and return here later for mixed retrieval.

Questions about Algebra II practice

When should I change the difficulty?

Move up after you can solve several questions accurately without hints and explain the method. Move down for one short set when errors show that a definition or setup is still uncertain.

How often should I practice?

Short sessions on several days usually build stronger recall than one long session. Revisit missed Algebra II skills the next day, then mix them with older topics later in the week.

Where can I review the lessons in order?

Use Algebra II practice by unit for a syllabus-aligned sequence with unit selection, practice, and a complete answer review.

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