ENEM Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in ENEM, not a summary of it.

1. 1

   Arithmetic

   4 topics

   • Natural Numbers
   • Integers
   • Rational Numbers
   • Real Numbers
2. 2

   Algebra

   4 topics

   • Polynomials
   • Equations and Inequalities
   • Functions
   • Progressions
3. 3

   Geometry

   4 topics

   • Plane Geometry
   • Solid Geometry
   • Analytic Geometry
   • Trigonometry
4. 4

   Statistics and Probability

   3 topics

   • Descriptive Statistics
   • Probability Theory
   • Combinatorics

Mathematics flashcards for ENEM

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. What is the set of natural numbers $\mathbb{N}$?

   The set of counting numbers $\mathbb{N} = \{0, 1, 2, 3, 4, \dots\}$ (including $0$ in the standard convention).

2. State the criterion of divisibility by $3$ for a natural number.

   A number is divisible by $3$ if and only if the sum of its digits is divisible by $3$.

3. How do you find the LCM of two numbers using their prime factorizations?

   Take every prime factor that appears in either number, each raised to the highest power in which it occurs, and multiply them: the LCM uses the greatest exponents.

4. What relationship links the GCD and LCM of two numbers $a$ and $b$?

   $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = a \cdot b$.

5. What defines the set of integers $\mathbb{Z}$?

   $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$, the natural numbers together with their negatives (and zero).

6. What is the absolute value $|x|$ of an integer $x$?

   $|x|$ is the distance of $x$ from $0$: $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$.

7. How is a rational number defined?

   A rational number is any number expressible as $\frac{p}{q}$ with $p, q \in \mathbb{Z}$ and $q \neq 0$; the set is denoted $\mathbb{Q}$.

8. How can you tell whether a fraction produces a terminating or repeating decimal?

   In lowest terms, the decimal terminates if the denominator's only prime factors are $2$ and/or $5$; otherwise the decimal is periodic (repeating).

9. Convert the repeating decimal $0.\overline{3}$ to a fraction.

   $0.\overline{3} = \frac{3}{9} = \frac{1}{3}$.

10. What distinguishes an irrational number, and how do the reals $\mathbb{R}$ relate to $\mathbb{Q}$?

    An irrational number cannot be written as $\frac{p}{q}$ (e.g. $\sqrt{2}, \pi$); its decimal is non-terminating and non-repeating. The reals are $\mathbb{R} = \mathbb{Q} \cup (\text{irrationals})$.

11. State the two main laws of exponents for products and powers of powers.

    $a^{m} \cdot a^{n} = a^{m+n}$ and $(a^{m})^{n} = a^{m \cdot n}$.

12. How do you rationalize the denominator of $\frac{1}{\sqrt{a}}$?

    Multiply numerator and denominator by $\sqrt{a}$: $\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$.

13. What is a monomial's degree and a polynomial's degree?

    A monomial's degree is the sum of the exponents of its variables; a polynomial's degree is the highest degree among its terms.

14. Give the expansion of the square of a binomial $(a+b)^{2}$.

    $(a+b)^{2} = a^{2} + 2ab + b^{2}$.

15. State the difference-of-squares factorization.

    $a^{2} - b^{2} = (a+b)(a-b)$.

16. Expand the cube of a binomial $(a+b)^{3}$.

    $(a+b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$.

17. Write the quadratic formula for solving $ax^{2}+bx+c=0$.

    $x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}$.

18. What does the discriminant $\Delta = b^{2}-4ac$ tell you about a quadratic's roots?

    If $\Delta > 0$ there are two distinct real roots; if $\Delta = 0$ one (double) real root; if $\Delta < 0$ no real roots (two complex roots).

19. State Vieta's relations for the roots $x_{1}, x_{2}$ of $ax^{2}+bx+c=0$.

    $x_{1}+x_{2} = -\dfrac{b}{a}$ and $x_{1} \cdot x_{2} = \dfrac{c}{a}$.

20. When solving an inequality, what operation forces you to reverse the inequality sign?

    Multiplying or dividing both sides by a negative number reverses the direction of the inequality (e.g. $-2x < 6 \Rightarrow x > -3$).

21. Define a function $f: A \to B$.

    A rule associating to each element $x \in A$ (domain) exactly one element $f(x) \in B$ (codomain).

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