Math · Unit 16

Precalculus

Polynomial functions, logarithms, sequences and complex numbers

Precalculus is the collection of tools calculus assumes you already have, gathered into one place.

It covers polynomial functions and their behaviour, logarithms and exponentials, trigonometric identities, sequences and series, and complex numbers.

This unit breaks down into 21 short steps and 120 questions, starting at difficulty 4 and building to 5. Below you can see exactly what it covers, how the path is structured, and worked examples with explanations.

Steps: 21
Questions: 120
Difficulty: 4-5

What this unit covers

Logarithms and Exponentials
Trigonometric Identities
Polynomial Functions
Sequences and Series
Complex Numbers

Where this fits

Needs Functions and Graphs and Geometry and Trigonometry. Straight into Calculus.

Where people slip

Logarithms are exponents rearranged, not a separate species. Every log identity is an exponent law written the other way round.

How the unit is structured

Precalculus runs as 21 short steps that unlock in order. 15 are practice rounds and 6 are challenge rounds that pull together everything before them. Questions start at difficulty 4 and climb to 5 as you progress.

Step 1 · easierStep 21 · harder

Challenge rounds

Example questions

30 real questions from this unit, with the answer and the reason behind it, grouped by what they practise. There are 120 in the unit altogether.

Complex Numbers

Build the sentenceLevel 4
1. Build a true statement about the imaginary unit.
Answer: Squaring the imaginary unit i gives negative one
Squaring i gives -1, which is what defines the imaginary unit.
Fact or fibLevel 4
2. The complex conjugate of 3 + 4i is 3 - 4i.
Answer: True
True, the conjugate keeps the real part and flips the sign of the imaginary part.
Fill the blankLevel 4
3. The modulus of the complex number 3 + 4i is ____.
- 5correct
- 7
- 25
- 12
The modulus is sqrt(3^2 + 4^2) = sqrt(25) = 5.
Multiple choiceLevel 4
4. What is i^2, where i is the imaginary unit?
- -1correct
- 1
- i
- -i
The imaginary unit is defined so that i^2 = -1.
Choose all that applyLevel 5
5. Select the true statements about the complex number 3 - 4i.
- Its conjugate is 3 + 4icorrect
- Its modulus is 5correct
- Its real part is 3correct
- It is a real number
Its conjugate is 3 + 4i, its modulus is sqrt(9 + 16) = 5, and its real part is 3; it is not real.
Match the pairsLevel 5
6. Match each power of i to its value.
Answer: i^2 = -1; i^3 = -i; i^4 = 1
i^2 = -1, i^3 = -i, and i^4 = 1.

Logarithms and Exponentials

Build the sentenceLevel 4
7. Build a true sentence about logarithms.
Answer: Logarithms are the inverse of exponential functions
Logarithms undo exponentials, so they are the inverse of exponential functions.
Fact or fibLevel 4
8. The natural logarithm, written ln, uses base e.
Answer: True
True, ln is the logarithm with base e, about 2.718.
Fill the blankLevel 4
9. For the same base, log(a) + log(b) = log(____).
- abcorrect
- a + b
- a / b
- a - b
The product rule for logarithms turns a sum of logs into the log of a product.
Match the pairsLevel 4
10. Match each logarithm to its value.
Answer: log base 2 of 4 = 2; log base 2 of 8 = 3; log base 2 of 16 = 4
Each value is the power of 2 that gives the number: 2^2 = 4, 2^3 = 8, 2^4 = 16.
Multiple choiceLevel 4
11. What is log base 2 of 8?
- 3correct
- 2
- 4
- 8
Since 2^3 = 8, the logarithm base 2 of 8 is 3.
Choose all that applyLevel 5
12. Select the true logarithm statements.
- ln(e) = 1correct
- log base 10 of 1 = 0correct
- log base 2 of 1/2 = -1correct
- log base 3 of 0 = 1
ln(e) = 1, log base 10 of 1 = 0, and log base 2 of 1/2 = -1 are true; log of 0 is undefined.

Polynomial Functions

Build the sentenceLevel 4
13. Build a true sentence about quadratic functions.
Answer: Quadratic functions produce graphs called parabolas
The graph of a quadratic function is a U-shaped curve called a parabola.
Choose all that applyLevel 4
14. Select all expressions that are polynomials.
- x^2 + 3x - 1correct
- 5x^3correct
- sqrt(x) + 1
- 2/x + 1
Polynomials use only whole-number exponents, so square roots and negative powers of x do not qualify.
Fact or fibLevel 4
15. A polynomial of degree 2 is called a quadratic.
Answer: True
True, degree 2 polynomials are quadratics.
Fill the blankLevel 4
16. A polynomial has exactly n complex ____ counting multiplicity, where n is its degree.
- rootscorrect
- turning points
- coefficients
- asymptotes
The Fundamental Theorem of Algebra guarantees n roots for a degree n polynomial.
Multiple choiceLevel 4
17. By the Fundamental Theorem of Algebra, how many complex roots (counting multiplicity) does a degree 3 polynomial have?
- 3correct
- 2
- 6
- 1
A polynomial of degree n has exactly n complex roots counting multiplicity, so degree 3 has 3.
Odd one outLevel 4
18. Which expression is the odd one out because it is not a polynomial?
- x^(1/2) + 3correct
- 4x^3 - x
- x^2 + 2x + 1
- 7
x^(1/2) + 3 has a fractional exponent, so it is not a polynomial; the others are.

Sequences and Series

Choose all that applyLevel 4
19. Select the sequences that are arithmetic.
- 2, 5, 8, 11
- 10, 7, 4, 1
- 1, 2, 4, 8
- 3, 3, 3, 3
Arithmetic sequences add a constant difference; 1, 2, 4, 8 doubles each time, so it is geometric.
Fill the blankLevel 4
20. The sum of a finite arithmetic series equals the number of terms times the ____ of the first and last term.
- averagecorrect
- product
- difference
- ratio
The sum is n times the average of the first and last term.
Guess the numberLevel 4
21. What is the sum 1 + 2 + 3 + ... + 100?
Answer: 5050
Using n/2 times (first + last) gives 100/2 times (1 + 100) = 5050.
Match the pairsLevel 4
22. Match each formula to what it computes.
Answer: Arithmetic nth term = a_1 + (n - 1)d; Geometric nth term = a_1 * r^(n - 1); Arithmetic sum = (n/2)(a_1 + a_n)
Arithmetic and geometric sequences have their own nth-term formulas, and the arithmetic sum uses (n/2)(a_1 + a_n).
Multiple choiceLevel 4
23. What is the sum 1 + 2 + 3 + ... + 10?
- 55correct
- 45
- 50
- 100
Using n/2 times (first + last) gives 10/2 times (1 + 10) = 55.
Fact or fibLevel 5
24. The infinite geometric series 1 + 2 + 4 + 8 + ... converges to a finite sum.
Answer: False
False, the ratio is 2, so the terms grow and the series diverges.

Trigonometric Identities

Choose all that applyLevel 4
25. Select the Pythagorean identities.
- sin^2(x) + cos^2(x) = 1correct
- 1 + tan^2(x) = sec^2(x)correct
- 1 + cot^2(x) = csc^2(x)correct
- sin(2x) = 2 sin(x) cos(x)
The three Pythagorean identities all come from sin^2 + cos^2 = 1; the double-angle formula is not one of them.
Fill the blankLevel 4
26. The identity 1 + tan^2(x) = ____ .
- sec^2(x)correct
- csc^2(x)
- cot^2(x)
- cos^2(x)
Dividing the Pythagorean identity by cos^2(x) gives 1 + tan^2(x) = sec^2(x).
Match the pairsLevel 4
27. Match each reciprocal trig function to its definition.
Answer: sec(x) = 1/cos(x); csc(x) = 1/sin(x); cot(x) = 1/tan(x)
Secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent.
Multiple choiceLevel 4
28. What does sin^2(x) + cos^2(x) equal for any angle x?
- 1correct
- 0
- 2
- sin(2x)
This is the Pythagorean identity, and it always equals 1.
Build the sentenceLevel 5
29. Build a true sentence defining tangent.
Answer: Tangent equals sine divided by cosine
Tangent equals sine divided by cosine.
Fact or fibLevel 5
30. The double-angle formula sin(2x) = 2 sin(x) cos(x) is correct.
Answer: True
True, that is the standard double-angle identity for sine.

Where these questions come from. Each unit starts as a plan of the concepts it should cover and the difficulty it should span. Questions are written against that plan with AI assistance, then checked by a validator that rejects anything without a single defensible answer, an explanation, or plausible wrong options. How we write questions sets out the whole process, and corrections are fixed in the bank and reach the site and the app the same day.

How you practise

This unit mixes 14 different question formats, so you are recalling and applying rather than recognising the same layout every time.

Build the sentence
Choose all that apply
Fact or fib
Fill the blank
Guess the number
Match the pairs
Multiple choice
Odd one out
Picture question
Put in order
Sort into groups
Spell it
True or false
Type the answer

Practise Precalculus

120 questions across 21 steps. Start with step one and crawl at your own pace.

Play this unit

Read about Precalculus

Explainers from our blog on what this unit covers. Each one ends with real questions from the bank.

More units in Math

See all 18 units in Math →