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Algebra Syllabus

Mathematical study and application. The full Algebra syllabus: 9 chapters and 51 topics, with 50 flashcards written against it.

9Chapters

51Topics

50Flashcards

Algebra syllabus — full chapter and topic list

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

Fundamentals of Algebra
4 topics
- Introduction to Variables
- Basic Algebraic Operations
- Order of Operations
- Properties of Real Numbers
Linear Equations and Inequalities
6 topics
- Solving Linear Equations
- Graphing Linear Equations
- Slope and Intercept
- Solving Linear Inequalities
- Systems of Linear Equations
- Applications of Linear Equations
Polynomials
6 topics
- Introduction to Polynomials
- Addition and Subtraction of Polynomials
- Multiplication of Polynomials
- Factoring Polynomials
- Special Products
- Division of Polynomials
Quadratic Equations
6 topics
- Introduction to Quadratic Equations
- Solving Quadratic Equations by Factoring
- Completing the Square
- The Quadratic Formula
- Graphing Quadratic Functions
- Applications of Quadratic Equations
Rational Expressions and Equations
5 topics
- Simplifying Rational Expressions
- Multiplication and Division of Rational Expressions
- Addition and Subtraction of Rational Expressions
- Solving Rational Equations
- Applications of Rational Expressions
Radicals and Exponents
6 topics
- Introduction to Exponents
- Laws of Exponents
- Simplifying Radical Expressions
- Operations with Radical Expressions
- Solving Radical Equations
- Complex Numbers
Functions and Their Graphs
6 topics
- Introduction to Functions
- Domain and Range
- Types of Functions
- Graphing Functions
- Transformations of Functions
- Inverse Functions
Exponential and Logarithmic Functions
6 topics
- Introduction to Exponential Functions
- Properties of Exponential Functions
- Introduction to Logarithmic Functions
- Properties of Logarithms
- Solving Exponential and Logarithmic Equations
- Applications of Exponential and Logarithmic Functions
Sequences, Series, and Probability
6 topics
- Arithmetic Sequences and Series
- Geometric Sequences and Series
- The Binomial Theorem
- Introduction to Probability
- Counting Principles
- Permutations and Combinations

Algebra flashcards

The complete Algebra deck: 50 questions with worked answers.

In algebra, what is a variable?
A variable is a symbol (usually a letter such as $x$ or $y$) that represents an unknown or changeable numerical value.
What is a coefficient in an algebraic term such as $7x$?
The coefficient is the numerical factor multiplying the variable. In $7x$, the coefficient is $7$.
What are like terms, and give an example?
Like terms have the same variables raised to the same powers; only their coefficients may differ. Example: $3x^{2}$ and $-5x^{2}$ are like terms.
What is a constant term in an algebraic expression?
A constant is a term with no variable; its value never changes, e.g., the $4$ in $2x + 4$.
State the order of operations (PEMDAS).
$$\text{Parentheses} \to \text{Exponents} \to \text{Multiplication/Division (left to right)} \to \text{Addition/Subtraction (left to right)}$$
Evaluate using order of operations: $3 + 4 \times 2^{2}$.
Exponent first: $2^{2}=4$; then multiply: $4\times4=16$; then add: $3+16=19$.
State the commutative properties of addition and multiplication.
$$a+b=b+a \qquad a\cdot b=b\cdot a$$
State the associative properties of addition and multiplication.
$$(a+b)+c=a+(b+c) \qquad (a\cdot b)\cdot c=a\cdot(b\cdot c)$$
State the distributive property.
$$a(b+c)=ab+ac$$
What are the additive identity and multiplicative identity?
The additive identity is $0$ (since $a+0=a$) and the multiplicative identity is $1$ (since $a\cdot 1=a$).
What is the additive inverse and the multiplicative inverse of a nonzero number $a$?
The additive inverse is $-a$ (since $a+(-a)=0$); the multiplicative inverse is $\frac{1}{a}$ (since $a\cdot\frac{1}{a}=1$).
What is the general form of a linear equation in one variable?
$$ax+b=0, \quad a\neq 0$$
Solve the linear equation $2x+5=13$.
Subtract $5$: $2x=8$; divide by $2$: $x=4$.
What operations keep an equation balanced when solving it?
Performing the same operation on both sides: adding, subtracting, multiplying, or dividing (by a nonzero number) both sides equally.
What is the slope-intercept form of a line?
$$y=mx+b$$ where $m$ is the slope and $b$ is the $y$-intercept.
What is the formula for the slope between two points $(x_1,y_1)$ and $(x_2,y_2)$?
$$m=\frac{y_2-y_1}{x_2-x_1}$$
What does the slope $m$ represent geometrically?
It is the rate of change, or $\frac{\text{rise}}{\text{run}}$ — the vertical change divided by the horizontal change along the line.
What is the slope of a horizontal line and of a vertical line?
A horizontal line has slope $m=0$; a vertical line has an undefined slope.
What is the $x$-intercept and $y$-intercept of a line?
The $x$-intercept is where the line crosses the $x$-axis ($y=0$); the $y$-intercept is where it crosses the $y$-axis ($x=0$).
What is the point-slope form of a linear equation?
$$y-y_1=m(x-x_1)$$

Show the other 30 Algebra flashcards

What is the standard form of a linear equation in two variables?
$$Ax+By=C$$ where $A$, $B$, and $C$ are constants.
How do the slopes of parallel and perpendicular lines relate?
Parallel lines have equal slopes ($m_1=m_2$); perpendicular lines have slopes that are negative reciprocals ($m_1\cdot m_2=-1$).
What key rule applies when solving a linear inequality by multiplying or dividing by a negative number?
You must reverse the direction of the inequality sign.
Solve the inequality $-3x>12$.
Divide by $-3$ and flip the sign: $x<-4$.
How is the solution to a linear inequality graphed on a number line for $<$/$>$ versus $\leq$/$\geq$?
Use an open circle for strict inequalities ($<$, $>$) and a closed/filled circle for inclusive inequalities ($\leq$, $\geq$).
What is a system of linear equations?
A set of two or more linear equations involving the same variables, whose solution is the set of values satisfying all equations simultaneously.
Name the three algebraic/graphical methods for solving a system of linear equations.
Substitution, elimination (addition), and graphing.
How many solutions can a system of two linear equations have, and what does each mean graphically?
One solution (lines intersect), no solution (parallel lines), or infinitely many solutions (same/coincident line).
In a system, what are the terms for consistent vs. inconsistent, and dependent vs. independent?
Consistent: at least one solution; inconsistent: no solution. Independent: exactly one solution; dependent: infinitely many solutions (same line).
Describe the substitution method for solving a system.
Solve one equation for one variable, substitute that expression into the other equation, solve for the remaining variable, then back-substitute.
Describe the elimination method for solving a system.
Add or subtract multiples of the equations so one variable cancels, solve for the remaining variable, then substitute back.
What is a monomial, binomial, and trinomial?
A monomial has one term, a binomial has two terms, and a trinomial has three terms.
What is the degree of a polynomial?
The highest exponent (sum of exponents in a term) among all terms; e.g., $4x^{3}+2x-1$ has degree $3$.
How do you add or subtract polynomials?
Combine like terms by adding or subtracting their coefficients; for subtraction, distribute the negative sign first.
Simplify: $(3x^{2}+2x-1)+(x^{2}-5x+4)$.
$$4x^{2}-3x+3$$
Simplify: $(5x^{2}-3x)-(2x^{2}+4x)$.
$$3x^{2}-7x$$
What is the product rule for exponents?
$$x^{m}\cdot x^{n}=x^{m+n}$$
What is the FOIL method used for?
Multiplying two binomials by summing the products of the First, Outer, Inner, and Last terms.
Expand $(x+3)(x+5)$ using FOIL.
$$x^{2}+8x+15$$
State the square of a binomial special products.
$$(a+b)^{2}=a^{2}+2ab+b^{2} \qquad (a-b)^{2}=a^{2}-2ab+b^{2}$$
State the difference of squares formula.
$$(a+b)(a-b)=a^{2}-b^{2}$$
Factor $x^{2}-9$.
$$(x+3)(x-3)$$
What is the greatest common factor (GCF) method of factoring?
Factor out the largest monomial dividing every term, e.g., $6x^{2}+9x=3x(2x+3)$.
Factor the trinomial $x^{2}+7x+12$.
$$(x+3)(x+4)$$
When dividing a polynomial by a monomial, what is the general approach?
Divide each term of the polynomial separately by the monomial, e.g., $\frac{6x^{3}+4x^{2}}{2x}=3x^{2}+2x$.
What algorithm is used to divide a polynomial by a binomial such as $(x-2)$?
Long division of polynomials (or synthetic division when dividing by a linear factor $x-c$).
What is the standard form of a quadratic equation?
$$ax^{2}+bx+c=0, \quad a\neq 0$$
State the Zero Product Property used in solving quadratics by factoring.
If $A\cdot B=0$, then $A=0$ or $B=0$ (at least one factor must equal zero).
Solve $x^{2}-5x+6=0$ by factoring.
Factor: $(x-2)(x-3)=0$, so $x=2$ or $x=3$.
What is a root (solution) of a quadratic equation, and how many can it have?
A root is a value of $x$ making the equation equal zero; a quadratic can have at most two real roots.

Algebra syllabus FAQ

What is in the Algebra syllabus?

The Algebra syllabus is split into 9 chapters: Fundamentals of Algebra, Linear Equations and Inequalities, Polynomials, Quadratic Equations, Rational Expressions and Equations, Radicals and Exponents, Functions and Their Graphs, Exponential and Logarithmic Functions and Sequences, Series, and Probability. Together they contain 51 topics.

How many chapters and topics are in the Algebra syllabus?

9 chapters and 51 topics. The largest chapter is Linear Equations and Inequalities, with 6 topics.

Are there Algebra flashcards?

Yes — 50 question-and-answer flashcards, printed in full on this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.

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