Algebra 2 help: all 7 units, with worked problems

Q: What are complex numbers in Algebra 2?

A complex number has the form a + bi, where i is the imaginary unit, the number whose square is -1. Complex numbers let you solve equations such as x^2 = -9, whose solutions are 3i and -3i.

What you’ll learn

The 7 units of Algebra 2

The course as it is organized in Mathflow, in order. Each unit is split into precise skills that you practice one at a time.

Unit 12 skills
Linear Systems & Matrices
Solve systems of three equations in three variables, and add, subtract and multiply matrices to represent and solve linear systems.
- Solve systems of linear equations in three variables
- Perform operations with matrices
Unit 22 skills
Complex Numbers
Add, subtract, multiply and divide complex numbers, using the conjugate to divide, and solve quadratic equations whose solutions are not real, in the form $a + bi$ .
- Perform operations with complex numbers
- Solve quadratic equations with complex solutions
Unit 33 skills
Polynomial Functions
Divide polynomials by long and synthetic division, use the Remainder and Factor Theorems to find zeros, and sketch graphs from end behavior and multiplicity.
- Divide polynomials using long and synthetic division
- Apply the Remainder and Factor Theorems
- Analyze and graph polynomial functions
Unit 43 skills
Rational & Radical Functions
Simplify and combine rational expressions, solve rational and radical equations while rejecting extraneous solutions, and graph rational functions with their asymptotes.
- Simplify and perform operations on rational expressions
- Solve rational and radical equations
- Graph rational functions and identify asymptotes
Unit 53 skills
Exponential & Logarithmic Functions
See logarithms as the inverse of exponents, use the product, quotient, power and change-of-base rules, and solve equations with $e$ and $\ln$ .
- Evaluate and use logarithms
- Apply properties of logarithms
- Solve exponential and logarithmic equations
Unit 62 skills
Trigonometric Functions
Read exact values of sine, cosine and tangent on the unit circle, and graph trigonometric functions with their amplitude, period, midline and shifts.
- Use the unit circle to find trigonometric values
- Graph trigonometric functions
Unit 72 skills
Statistics & Probability
Use the mean and standard deviation of a normal distribution to estimate percentages and $z$ -scores, and judge surveys, observational studies and experiments.
- Analyze data using the normal distribution
- Evaluate statistical experiments and sampling

Algebra 2 at a glance

In the traditional US sequence, Algebra 2 is the third course, after Algebra 1 and Geometry. Many students take it in 11th grade.

Before you start. You need the factoring, quadratic functions and exponential models of Algebra 1, and the right-triangle trigonometry of Geometry.

After Algebra 2. Pre-Calculus takes these functions further, with polar functions, vectors and matrices, and prepares you for Calculus.

In Mathflow. The course has $7$ units and $17$ skills. Each skill is a series in three levels, and each unit ends with a Chapter Challenge.

Worked problems

Two problems, solved step by step

The zeros of a polynomial and an exponential equation: two classics of Algebra 2. Try each one on paper before you read the solution.

Problem 1
All the zeros of a cubic
Find every zero of $p(x) = 2x^3 - 3x^2 - 11x + 6$ , then write $p(x)$ in factored form.
Solution
By the Rational Root Theorem, a rational zero has the form $\dfrac{\text{factor of } 6}{\text{factor of } 2}$ . The candidates are $\pm 1$ , $\pm 2$ , $\pm 3$ , $\pm 6$ , $\pm \dfrac{1}{2}$ and $\pm \dfrac{3}{2}$ .
Test them. $p(1) = -6$ and $p(2) = -12$ , but $p(3) = 54 - 27 - 33 + 6 = 0$ . By the Factor Theorem, $x - 3$ is a factor.
Divide by $x - 3$ with synthetic division. The coefficients $2, -3, -11, 6$ give $2, 3, -2$ with a remainder of $0$ :
$p(x) = (x - 3)(2x^2 + 3x - 2)$
Factor the quadratic: $2x^2 + 3x - 2 = (2x - 1)(x + 2)$ . So
$p(x) = (x - 3)(2x - 1)(x + 2)$
The zeros are $3$ , $\dfrac{1}{2}$ and $-2$ . Check: $p(-2) = -16 - 12 + 22 + 6 = 0$ .
Problem 2
When does the account reach $3,000?
You deposit $2,000 in an account that earns $5\%$ interest, compounded once a year. After how many years will the balance first reach $3,000?
Solution
After $t$ years, the balance is $2000 \cdot 1.05^t$ dollars. Set it equal to $3000$ and divide both sides by $2000$ :
$1.05^t = 1.5$
Take the natural logarithm of both sides and use the power rule, $\ln(a^t) = t \ln a$ :
$t = \dfrac{\ln 1.5}{\ln 1.05} \approx 8.31$
Interest is added once a year, so the balance only changes at the end of each year. After $8$ years it is about $2,954.91, and after $9$ years about $3,102.66. The balance first reaches $3,000 after $9$ years.

How Mathflow teaches it

Algebra 2 in three levels

Every skill of the course is a series of short missions in three levels. Here is what you actually do in three of them.

Level 1 · Explore
Explore and play with the concepts to build your intuition.
Level 2 · Explain
Turn your intuitions into the right words and uncover hidden mistakes.
Level 3 · Solve
Reason it out, crunch the numbers, and find your way to the answer.

You don’t start with a list of rules. In the first level, you act on the math: you move a parabola, drag a curve, add half-lives and watch the solutions change. In the second level, you put words on what you saw: you decide which of two students is right, find the first wrong line of a solution, or complete a sentence that explains a method. In the third level, you solve problems on your own, like the ones you meet in class and on tests.

If you get stuck, Alphi, the AI math tutor, gives you a hint instead of the answer, at any hour. When the skills of a unit are done, a Chapter Challenge checks the whole unit.

Unit 2 · Complex Numbers
1Solve quadratic equations with complex solutions
- Level 1 · Explore
  Change $n$ until the solutions of $x^2 = n$ are $\pm 4i$ . Then drag the vertex of a parabola until the solutions in the box are $1 \pm 2i$ .
- Level 2 · Explain
  Lena and Omar both rewrote $x^2 + 4x + 13 = 0$ as $(x + 2)^2 = -9$ , and you decide who is right. You also tap the first wrong line of Ethan’s quadratic formula on $2x^2 - 4x + 10 = 0$ .
- Level 3 · Solve
  One solution of $x^2 + bx + c = 0$ is $-2 + 5i$ , with $b$ and $c$ real. You find $b$ and $c$ .
Unit 3 · Polynomial Functions
2Apply the Remainder and Factor Theorems
- Level 1 · Explore
  Drag a curve up or down until $x - 2$ is a factor of $P(x)$ , and tap every binomial that is a factor of $P(x) = x^3 - 4x^2 + x + 6$ .
- Level 2 · Explain
  Kayla tests $P(x) = x^3 + 2x^2 - 5x - 6$ , and you tap the first wrong line of her work. You also use $P(1) = 0$ to find every zero of $x^3 - 7x + 6$ .
- Level 3 · Solve
  Dividing $x^3 + kx - 6$ by $x - 2$ leaves the remainder $4$ . You find the number $k$ .
Unit 5 · Exponential & Logarithmic Functions
3Solve exponential and logarithmic equations
- Level 1 · Explore
  A $96$ -gram sample has a half-life of $8$ days: add half-lives until $12$ grams remain. Then isolate $x$ in $5 \cdot 3^{x-2} = 45$ with buttons that act on both sides.
- Level 2 · Explain
  Ryan solved $\log_2(x + 3) = 5$ , and you find what is wrong with his answer. You also sort values into solutions and extraneous solutions.
- Level 3 · Solve
  A car’s value is $V(t) = 20{,}000(0.85)^t$ dollars after $t$ years. You find when it is worth $10,000, to the nearest tenth of a year.

Practice Algebra 2 for free

Common mistakes

Five mistakes to avoid

These slips come up again and again in Algebra 2. Learn to spot them, and you’ll catch them in your own work.

Multiplying square roots of negative numbers
The rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails when $a$ and $b$ are both negative. Write each root with $i$ first: $\sqrt{-4} \cdot \sqrt{-9} = 2i \cdot 3i = 6i^2 = -6$ , not $\sqrt{36} = 6$ .
Splitting the log of a sum
$\log(x + y)$ is not $\log x + \log y$ : the product rule is $\log(xy) = \log x + \log y$ . For example, $\log_2(4 + 4) = \log_2 8 = 3$ , while $\log_2 4 + \log_2 4 = 4$ .
Skipping the check for extraneous solutions
Squaring both sides can add solutions. Squaring $\sqrt{x + 2} = x$ gives $x^2 - x - 2 = 0$ , so $x = 2$ or $x = -1$ . But $\sqrt{-1 + 2} = 1$ , not $-1$ : only $x = 2$ is a solution.
Canceling terms instead of factors
In $\dfrac{x^2 - 9}{x - 3}$ , you can’t cancel the $x$ ’s or the $3$ ’s. Factor first: $\dfrac{(x - 3)(x + 3)}{x - 3} = x + 3$ , for $x \neq 3$ .
Taking $b$ for the period
In $y = \sin(bx)$ with $b > 0$ , the period is $\dfrac{2\pi}{b}$ , not $b$ . So $y = \sin(2x)$ repeats every $\pi$ , not every $2$ .

Frequently asked questions

What do you learn in Algebra 2?

Systems in three variables and matrices, complex numbers, polynomial, rational and radical functions, exponential and logarithmic functions, trigonometric functions, and statistics with the normal distribution. In Mathflow, that’s 7 units and 17 skills.

What grade is Algebra 2?

In the traditional US sequence, Algebra 2 comes after Algebra 1 and Geometry. Many students take it in 11th grade.

Is Algebra 2 harder than Algebra 1?

It goes further: more kinds of functions, more steps per problem, and new numbers such as $i$ . It builds directly on Algebra 1, so solid factoring and equation solving make a big difference.

What are complex numbers in Algebra 2?

A complex number has the form $a + bi$ , where $i$ is the imaginary unit, the number whose square is $-1$ . Complex numbers let you solve equations such as $x^2 = -9$ , whose solutions are $3i$ and $-3i$ .

What comes after Algebra 2?

Usually Pre-Calculus, which takes functions and trigonometry further and adds polar functions, vectors and matrices. Calculus comes after that.

How can I practice Algebra 2 at home?

Redo the worked problems on this page without looking, then practice skill by skill. Mathflow covers the whole course in three levels per skill, with Alphi, an AI math tutor, for hints when you’re stuck. It’s free to start on iOS and Android.

Start for free

Standards

Algebra 2 and the Common Core

The Common Core State Standards organize high school math by conceptual categories, not by course. An Algebra 2 course draws on Number and Quantity (the complex number system), Algebra (polynomials and rational expressions, equations), Functions (building and interpreting functions, exponential models, trigonometric functions) and Statistics and Probability.

Appendix A of the standards proposes a traditional US pathway of three courses, Algebra I, Geometry and Algebra II. Its Algebra II course includes complex numbers, polynomial and rational expressions, logarithms as solutions of exponential equations, trigonometric functions on the unit circle, and statistical inference. Many states use their own version of the standards, so check your school’s course outline for the exact list.

Algebra 2, explained unit by unit

The 7 units of Algebra 2

Linear Systems & Matrices

Complex Numbers

Polynomial Functions

Rational & Radical Functions

Exponential & Logarithmic Functions

Trigonometric Functions

Statistics & Probability

Algebra 2 at a glance

Two problems, solved step by step

All the zeros of a cubic

When does the account reach $3,000?

Algebra 2 in three levels

Level 1 · Explore

Level 2 · Explain

Level 3 · Solve

1Solve quadratic equations with complex solutions

2Apply the Remainder and Factor Theorems

3Solve exponential and logarithmic equations

Five mistakes to avoid

Multiplying square roots of negative numbers

Splitting the log of a sum

Skipping the check for extraneous solutions

Canceling terms instead of factors

Taking $b$ for the period

Frequently asked questions

Algebra 2 and the Common Core

Math is all about flow.

Algebra 2, explained unit by unit

Linear Systems & Matrices

Complex Numbers

Polynomial Functions

Rational & Radical Functions

Exponential & Logarithmic Functions

Trigonometric Functions

Statistics & Probability

Algebra 2 at a glance

All the zeros of a cubic

When does the account reach $3,000?

Level 1 · Explore

Level 2 · Explain

Level 3 · Solve

1Solve quadratic equations with complex solutions

2Apply the Remainder and Factor Theorems

3Solve exponential and logarithmic equations

Multiplying square roots of negative numbers

Splitting the log of a sum

Skipping the check for extraneous solutions

Canceling terms instead of factors

Taking bbb for the period

Math is all about flow.

Taking $b$ for the period